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LOS  ANGiiLEb 
LIBRARY 


INTRODUCTION  TO 
ECONOMIC  STATISTICS 


INTRODUCTION    TO 

ECONOMIC  STATISTICS 


BY 
GEORGE  R.  DAVIES,  Ph.  D. 

PROFESSOR  OF  SOCIOLOGY,  UNIVERSITY  OF  NORTH  DAKOTA 


tWi»f^<jj*W\i**t 


NEW  YORK 

THE  CENTURY  CO. 

1922 

91409 


Copyright,  1922,  by 
The  Century  Co. 


Printed  in  U.  S.  A. 


"K 


.%;: 


.    .5 


,  PREFACE 

'  The  application   of  statistical  methods  to   special 

fields  such  as  demography,  education,  and  economics 
has  in  recent  years  advanced  very  rapidly.  As  a  re- 
sult a  study  of  statistical  principles  must  be  confined 
chiefly  to  one  of  the  special  fields  if  it  is  not  to  be 
lost  in  the  multiplicity  of  specific  methods  and  illus- 
trations.   This  text-book  has  been  written  with  the  in- 

►i      terests  of  the  student  of  economics  in  mind. 

A  common  difficulty  which  the  teacher  of  statistics 
encounters  is  a  lack  of  provision  for  laboratory  work. 
An  attempt  has  here  been  made  to  supply  this  need 
by  furnishing  illustrative  problems,  graphs,  and  data 
which  may  be  worked  over  by  the  student,  and  by  add- 

1^5]      ing  to  each  chapter  a  list  of  related  exercises.     The 

^  exercises  will  be  found  extensive  enough  so  that  the 
teacher  may  select  those  which  are  adapted  to  his  re- 
quirements. The  longer  problems  should  be  subdivided, 
and  the  parts  assigned  to  different  members  of  the 
class.  Both  the  tables  and  the  exercises  may  very  well 
^     be  supplemented  by  the  use  of  data  drawn  from  such 

^  sources  as  the  Survey  of  Current  Business,  the  Monthly 
Labor  Review,  and  the  Statistical  Abstract  of  the 
United  States  (Superintendent  of  Documents,  Govern- 
ment Printing  Office,  Washington,  D.  C).  The  topics 
covered  in  the  text  represent  probably  a  maximum  of 
what  can  be  mastered  by  a  college  class  in  a  term. 
Perhaps  it  may  be  found  advisable  to  omit  certain 
topics,    such  as  interpolating  for  quartiles,  theories 


vi  PREFACE 

of  price  indexes,  parabola  trends,  seasonal  variations, 
and  the  more  complex  methods  of  correlation. 

Nearly  all  the  material  here  presented  has  been  ac- 
cumulated from  experience  in  the  statistical  laboratory 
and  class-room.  Particular  attention  has  been  given 
to  the  requirements  in  respect  to  fundamental  theory 
of  the  statistical  departments  of  the  larger  banks  and 
business  houses.  Some  of  the  recently  developed 
methods  of  handling  business  barometers  have  there- 
fore been  touched  upon,  and  some  attention  has  been 
given  to  the  theory  of  price  and  production  indexes. 

The  book  is  an  outgrowth  of  the  undergraduate 
course  in  statistics  given  by  the  writer  at  Princeton 
University  during  the  school  year  1920-1921.  This 
course  was  modeled  in  its  general  features  upon  the 
course  given  the  preceding  year  by  Professor  J.  H. 
Williams,  now  of  Harvard  University.  The  writer 
wishes  to  acknowledge  his  indebtedness  to  Professor 
Williams  for  the  general  plan  of  the  laboratory  exer- 
cises, as  well  as  for  many  specific  suggestions.  Thanks 
are  also  extended  to  Professors  F.  A.  Fetter  and  E. 
W.  Kemmerer  of  Princeton  for  their  interest  and  en- 
couragement, and  to  Professor  W.  F.  Willcox  of  Cor- 
nell who  read  the  first  draft  of  the  manuscript  and 
made  several  valuable  suggestions  for  its  revision.  In- 
debtedness is  also  acknowledged  to  the  following  for 
their  kind  permission  to  reprint  data:  Mr.  Roger  W. 
Babson,  Professor  Stanley  E.  Howard,  Bureau  of  La- 
bor Statistics,  National  City  Bank,  National  Bureau  of 
Economic  Research,  National  Industrial  Conference 
Board,  Review  of  Economic  Statistics,  and  the  Quar- 
terly Journal  of  the  University  of  North  Dakota. 


CONTENTS 

CHAPTER  PAGE 

I    Tabulation 3 

II    Types  and  Measures  of  Dispersion      ....  20 

III  Indexes  of  Wages  and  Prices  ......  47 

IV  Quantity  Indexes  and  Their  Uses  .....  74 
V     Trends  and  Cycles 100 

VI     Correlation 131 

Appendix  I 

Laboratory  Material  and  References  .     .     .  153 

Appendix  II 

Tables  of  Powers  and  Roots 157 

Appendix  III 
A  Picture  of   the  Progress  of   the  United 

States  During  120  Years  of  National  Life  .  160 

Index 161 


INTRODUCTION  TO 
ECONOMIC  STATISTICS 


INTRODUCTION  TO  ECONOMIC 
STATISTICS 

CHAPTER  I 

TABULATION 

The  term  ** statistics"  when  used  to  designate  a 
branch  of  study,  implies  an  exposition  of  certain 
methods  employed  in  presenting  and  interpreting  the 
numerical  aspects  of  a  given  subject.  The  science  of 
statistics  consists,  therefore,  of  principles  and  methods, 
rather  than  of  data.  The  principles  are  essentially 
the  same  whether  the  application  is  made  to  biology, 
demography,  education,  or  economics.  But  the  de- 
tailed methods  in  these  and  other  fields  have  in  late 
years  become  so  specialized  that  it  is  hardly  practi- 
cable any  longer  to  study  statistics  in  the  abstract. 
The  field  of  application  here  adopted  is  chiefly  that  of 
general  economics.^  Illustrations  will  be  given  of  the 
methods  employed  in  organizing  data,  in  computing 
and  employing  indexes,  and  in  measuring  trends  and 
correlations. 

*  It  should  be  noted  that  economic  statistics  are  commonly  distin- 
guished from  business  statistics.  The  former  subject  studies  general 
market  conditions,  while  the  latter  subject  deals  with  the  details  of  a 
specific  business  establishment,  and  is  therefore  an  adjunct  of  account- 
ing. Business  statistics  vary  so  greatly  from  one  establishment  to  an- 
other that  it  is  difficult  to  generalize  from  them.  Their  problems 
consist  largely  of  the  application  of  statistical  principles  to  specific 
situations.  For  a  discussion  of  the  distinction  see  "The  Scope  of  Busi- 
ness Statistics,"  by  R.  P.  Falkner,  in  the  Quarterly  Publications  of 
the  American  Statistical  Association,  June,  1918,  pp.  24-29. 

3 


4     INTRODUCTION  TO  ECONOMIC  STATISTICS 

Preliminary  Schedules.  Statistical  field  work  com- 
monly begins  with  the  preparation  of  schedules  in 
which  to  enter  the  desired  data.  These  schedules  may 
be  in  the  nature  of  questionnaires,  or  they  may  be  mere 
forms  in  w^hich  to  copy  certain  records.  Since  such 
schedules  will  vary  with  the  particular  task  in  hand, 
few  rules  can  be  laid  down  for  their  preparation.  Ex- 
perience will  teach,  however,  that  they  must  be  very 
carefully  worked  out  in  advance.  In  the  first  place,  it 
must  be  determined  as  precisely  as  possible  what  data 
will  be  needed.  If  the  schedule  is  in  questionnaire 
form,  great  care  must  be  taken  to  make  the  questions 
unambiguous  and  easily  comprehended  by  those  who 
are  to  answer.  Errors  may  often  be  checked  by  asking 
a  question  in  two  ways,  at  different  places  in  the  list ; 
as  by  calling  for  both  the  age  and  the  date  of  birth. 

After  the  preliminary  schedules  have  been  compiled, 
the  statistician  begins  to  organize  his  data,  so  that  con- 
clusions may  be  deduced  and  presented  in  simplified 
form.  In  so  doing  he  will  make  use  of  the  processes 
of  tabulation.  These  may  best  be  explained  by  taking 
an  example.  As  such  an  example,  we  shall  choose 
certain  wage  schedules  which  have  been  recorded  in 
the  Aldrich  Report  on  ''Wholesale  Prices,  Wages,  and 
Transportation"  (Senate  Report  No.  1394,  dated 
1893).  The  data  here  used  will  be  found  in  Vol.  IV, 
pages  1463-1497,  and  refer  to  a  Connecticut  woolen 
mill  designated  as  Establishment  No.  86.  The  wage 
rolls  as  recorded  in  the  report  cover  about  half  a 
century,  ending  in  1891.  They  show  the  daily  wages 
paid  to  each  class  of  workmen  employed  in  the  mill 
in  January  and  July  of  each  year.    We  shall  select 


TABULATION  5 

for  tabulation  only  the  wages  paid  in  July  of  1870, 
1880,  and  1891. 

The  Primary  or  General  Purpose  Table.  In  tran- 
scribing the  selected  wage  schedules  into  a  primary 
table,  it  will  be  found  advisable  to  edit  the  original 
figures  by  making  certain  minor  modifications.  An  in- 
spection of  the  schedules  will  show  that  most  of  the 
wage  rates  are  expressed  as  multiples  of  five  cents. 
Of  course,  we  may  assume  theoretically  that  the  exact 
economic  values  which  are  approximated  in  the  actual 
wages  must  form  a  continuous  series,  instead  of  one 
having  a  regular  interval.  That  is,  if  the  wages  could 
be  expressed  as  theoretically  exact  values,  and  if  a 
very  large  number  of  workers  were  involved,  the  rates 
would  be  separated  by  intervals  of  only  a  small  frac- 
tion, of  a  cent.  The  case  may  be  compared  with  the 
measurement  of  the  height  of  the  individuals  in  a 
large  group.  If  the  measurements  are  taken  with 
very  precise  instruments,  the  results  will  be  expressed 
in  hundredths  or  perhaps  thousandths  of  an  inch.  But 
for  practical  purposes  measurements  to  perhaps  the 
nearest  quarter  inch  are  sufficiently  accurate.  In  the 
same  way  when  the  employer  made  an  offer  of  wages  he 
set  his  figure  at  a  multiple  of  five  cents,  or  in  the  case 
of  the  larger  wages  at  a  multiple  of  twenty-five  cents. 
For  his  purpose,  such  an  estimate  of  the  market  value 
of  labor  was  sufficiently  accurate.  The  exceptions  will 
be  found  to  be  rates  that  were  paid  to  an  especially 
large  number  of  workers.  In  such  cases  differences 
of  a  cent  or  two  are  of  considerable  consequence. 

Continuous  ajid  Discrete  Series.  A  series  of  meas- 
urements or  values  occurring  only  at  more  or  less  reg- 


6     INTRODUCTION  TO  ECONOMIC  STATISTICS 

ular  intervals  is  said  to  be  discrete.  Sometimes  a  series 
A\all  be  naturally  discrete,  as  when  flowers  are  clas- 
sitied  by  the  number  of  petals,  or  when  the  spots  in 
a  number  of  dice-throws  are  tabulated.  But  a  series 
which  is  theoretically  continuous  becomes  artificially 
discrete  when  a  limit  of  accuracy  is  determined  upon, 
as  when  height  is  measured  to  the  nearest  quarter- 
inch,  or  wages  are  expressed  in  multiples  of  five  cents. 
Our  series  of  wage  rates,  as  it  stands  in  the  records, 
is  therefore  discrete  at  intervals  of  five  cents,  except 
for  a  few  items. 

For  purposes  of  classification  it  is  desirable  to  have 
our  series  of  wage  rates  regularly  discrete  throughout. 
Since  it  is  not  possible  to  break  down  the  five  cent 
intervals  to  smaller  ones,  it  will  be  necessary  to  modify 
such  rates  as  67c.  and  $1.28  so  as  to  classify  them  as 
multiples  of  five.  If  there  were  only  a  few  scattered 
cases  .of  this  sort  affecting  only  a  small  number  of 
workers,  they  might  merely  be  entered  at  the  nearest 
five  cent  intervals;  that  is,  67c.  could  be  entered  as 
65c.  and  $1.28  as  $1.30.  But  since  the  number  of  work- 
ers at  these  irregular  rates  is  exceptionally  large, 
such  a  procedure  might  result  in  too  great  a  degree  of 
inaccuracy.  We  shall  therefore  apply  a  familiar  arith- 
metical device  and  break  up  the  given  number  of  work- 
ers at  each  inconvenient  rate  into  two  groups,  one 
having  a  higher  and  one  a  lower  rate  than  that  stated, 
but  maintaining  together  the  same  average  wage.  The 
procedure  may  be  illustrated  as  follows : 

{9  workers  at  65c. 
-f 
6  workers  at  70c. 


TABULATION 


This  result  is  obtained  by  taking  3/5  and  2/5  of  the 
fifteen  workers  (the  fractional  parts  of  the  five  cent 
interval  which  lie  between  67c.  and  the  nearest  mul- 
tiples of  5)  and  placing  the  larger  of  the  two  results 
at  the  rate  expressed  by  the  multiple  of  five  nearest  to 
67c.,  that  is,  at  65c.    The  smaller  of  the  two  results  is 

TABLE  I 

WAGE  EOLL  IN  A  CONNECTICUT  WOOLEN  MILL 
JULY  CTF  SPECIFIED  YEARS 


OCCUPATION 


Burlers 

Card  cleaners. 

Card  tenders. . 


Carpenters 

Cloth  Inspectors. 
Drawers-in 


Dressers. 


Djera. 


Firemen 

Foremen-burlers . . . . 
Fullers  and  giggers. 


Hand«ra-in .... 
Harness  hands. 
Loom  fixers. . . 


1870 


NO.       WAfiB 


$  .80 

.85 

.70 

1.10 

1.25 

.55 
.60 
.65 
.80 
2,75 

1.25 


1880 


NO.       WAGE 


1.35 
1.50 


1.50 

1.10 
1.15 
1.25 
1.35 
1.50 
.40 

1.50 
1.10 
1.15 


11* 


1 
1 

1* 
1* 

2 
2 


^  .80 

.85 

1.15 


.60 
.65 


2.75 
1.60 
1.90 
1.95 

1.50 
1.55 


1.25 
1.30 


1.05 
1.20 
1.25 


.40 
1.95 


1891 


NO.       WAGE 


14^ 


$1.10 


1 

1.10 

1 

1.15 

1 

1.20 

2 

1.25 

1 

.75 

1 

.85 

1 

.90 

1 

1.00 

1 

2.75 

1 

1.90 

1* 

1.75 

1* 

1.85 

4* 

1.90 

1 

1.25 

2 

1.60 

1 

1.65 

3 

1.75 

1 

1.15 

13 

1.25 

1 

1.40 

2 

1.50 

3 

1.50 

1 

2.25 

2 

1.05 

4 

1.10 

3 

1.15 

3 

1.25 

1 

1.50 

5» 

.50 

1 

1.50 

1 

2.00 

2 

2.10 

4 

2.20 

8     INTRODUCTION  TO  ECONOMIC  STATISTICS 


TABLE  I    (Continued) 


OCCUPATION 


Machinists 

Machinists — helpers 

Number  sewers 

Overseers — Carding  dept .  . 

"  Dyehouse 

"  Finishing  dept. 

*  *  Fulling  dept .  . . 

"  Spinning  dept. 

"  Spooling  dep't.  , 

**            Weaving  dept. 
Piecers 


Second  hands. 
Sewers 


Shearers . 


Sorters. . 
Speckers. 


Spinners — jack  and  mule. 
Spoolers 


Teamsters. 
Twisters .  . 
Watchmen . 


Weavers . 


Weavers — pattern . 


Winders 

Yam  carriers 
Totals.  . . 


1870 


NO.      WAGE 


1 
1 

1 

2 
1 

1* 

3 

7 


4* 
3* 
19 
5 


1^ 
1 
108 


^2.75 
1.75 

3.50 
3.75 
2.50 
2.75 
2.75 
2.25 
3.00 
.75 
.80 
1.50 
1.75 


1.15 
1.40 
1.50 
2.00 
2.75 
1.35 

1.75 

1.80 

.60 

1.50 

1.50 


1.05 
1.10 
1.30 
1.35 


1.00 
1.25 


1880 


NO.      WAGE 


1 
1 
1 
1 
1 
1 
1 

2 
8 
1 
1 
1* 


2 
1 

1* 
2* 
1 

5* 

4* 

1 

7* 

1 


79 
10' 

2' 


213 


1891 


$3.00 


3.25 
3.25 
3.00 
2.50 
3.00 
2.25 
3.00 
.70 
.75 
1.25 
1.50 
1.25 

1.25 


1.80 

2.75 

.90 

.95 

1.50 

.65 

.70 
1.50 

.90 
1.50 


1.20 
1.40 
1.45 


.95 
1.00 
1.75 


2 
1 
2* 
2* 
4 
7 
9* 
14* 
1 

4* 
1 
1 
1 


89 

60 

3 

2 

4 

1 

1 

12* 

17' 

1 

361 


NO.      WAGE 


$2.75 
2.10 
.90 
4.00 
4.25 
3.50 
2.50 
2.75 
2.50 
3.00 


1.75 

1.00 
1.25 
1.35 


1.70 
2.75 
.80 
.85 
1.25 
1.30 
.75 
.80 
1.60 
.90 
1.30 
1.35 
1.40 
1.30 
1.35 
1.70 
1.75 
1.25 
1.35 
1.50 
1.75 
2.00 
1.15 
1.20 
1.85 


Female  employes. 


TABULATION  9 

placed  at  the  rate  of  70c.  That  the  average  wage  has 
not  been  changed  by  this  operation  is  shown  by  the  fact 
that 

15  X  67c.  =  9  X  65c  +  6  X  70c. 

By  the  foregoing  method  all  irregular  wage  rates 
may  now  be  reduced  to  approximately  equivalent  mul- 
tiples of  five  cents.  Of  course,  when  a  fraction  appears 
in  the  operation  the  nearest  whole  number  is  taken. 
Thus  modified,  our  wage  schedules  appear  as  shown 
in  Table  I,  which  may  be  taken  as  an  example  of  a 
primary  or  general  purpose  table. 

The  Frequency  Curve.  The  tabulation  which  we  are 
about  to  undertake  has  for  its  immediate  object  the 
presentation  of  the  frequency  distribution  of  the  wages 
in  question.  Since  the  concept  of  a  frequency  distribu- 
tion has  a  concise  theoretical  basis,  it  will  be  of  ad- 
vantage to  turn  briefly  at  this  point  to  the  theoretical 
aspects  of  the  subject. 

If  we  take  the  square  of  a  binomial,  as  ar  -f  2ab  -{-  b-, 
we  have  three  classes  of  values  as  expressed  by  the 
letters  and  their  exponents,  and  these  classes  have 
frequencies  expressed  by  their  coefficients,  1:2:1.  If 
instead  of  the  second  power  we  take  the  fourth  power 
of  the  binomial,  we  have  five  classes  of  values,  having 
frequencies  respectively  of  1 :4 :6 :4 :1.  These  frequen- 
cies graphed  as  vertical  blocks  will  form  a  figure  such 
as  is  outlined  by  the  dotted  line  in  Fig.  1.  If  instead 
of  the  fourth  power  of  the  binomial  we  should  take  the 
thousandth  or  millionth  power,  the  steps  in  this  blocked 
frequency  polygon  would  practically  disappear,  and 
the  figure  would  approach  a  smoothed  bell-shaped 
curve  as  indicated  in  the  same  figure.   This  theoretical 


10  INTRODUCTION  TO  ECONOMIC  STATISTICS 

distribution  of  classes  of  values,  or  something  similar 
to  it,  may  be  discovered  to  exist  very  generally  in 
natural  and  social  phenomena,  and  is  also  the  expres- 
sion of  what  are  known  as  the  laws  of  chance.  The 
length  of  leaves  on  a  given  tree,  the  height  of  a  group 
of  persons,  the  per  cent  net  earnings  of  corporations, 


IOOa 


Figure  1.  NormEil  frequency  curve  (solid  line),  and  an  approxima- 
tion to  it  (dotted  line)  based  on  the  fourth  power  of  a  bionomial. 
Horizontal  scale  in  units  of  standard  deviation  from  average  (0). 
Quartilo  deviation  (Q.D.),  and  average  deviation   (A.D.) 

or  the  deviations  from  normal  of  a  price  index  through 
a  series  of  years,  will  show  when  properly  classified 
and  graphed  an  approximation  to  the  bell-shaped  fre- 
quency curve.  In  order  to  discover  whether  this  curve 
is  inherent  in  a  given  set  of  data,  it  is  necessary  first 
that  the  data  contain  a  considerable  number  of  items, 
and  second  that  the  classification  be  suitably  adjusted 
to  the  range  and  numbers.  If,  for  example,  the  height 
of  a  hundred  persons  were  taken  merely  to  the  nearest 


TABULATION  11 

foot,  only  two  or  three  classes  would  appear.  If,  how^ 
ever,  the  measurements  were  taken  accurately  to  .01 
inch,  so  many  classes  would  appear  that  the  fre- 
quencies would  be  hopelessly  scattered.  But  if  we 
made  our  measurements  to  the  nearest  inch,  we  would 
obtain  a  series  of  frequencies  somewhat  like  the  fol- 
lowing (the  classes  ranging  from  60  to  73  inches  in- 
clusive):  1:2:4:7:10:14:16:16:12:8:5:3:1:1.  These  fre- 
quencies when  graphed  will  give  an  approximation  to 
the  bell-shaped  curve.  It  will  be  necessary,  therefore, 
in  tabulating  our  wage  data  to  work  out  experimentally 
the  most  suitable  classification. 

The  Tally  Sheet  and  Frequency  Table.  To  facilitate 
the  classification  of  the  wages  selected  for  study,  a  tally 
sheet  is  drawn  up  as  shown  in  the  first  two  columns  of 
Table  II.  We  shall  show  here  the  details  for  only  the 
1891  figures,  leaving  the  1870  and  1880  figures  to  be 
worked  out  by  the*student.  In  studies  where  the  items 
must  be  entered  singly,  it  is  customary  to  use  the  famil- 
iar ''four  and  cross"  method  of  tallying  (/>y/  =5), 
but  this  is  inappropriate  when  the  items  are  already 
partially  grouped  as  they  are  here.  In  this  case  the 
number  of  workers  as  shown  in  the  wage  roll  is  entered 
in  the  appropriate  line  of  the  tally  sheet,  much  as  jour- 
nal items  are  posted  to  a  ledger.  Each  entry  is  sepa- 
rated from  adjacent  ones  by  a  dash.  Each  line  is  then 
totaled,  and  the  result  entered  under  the  five  cent 
column  of  ''Frequency  Classes."  A  series  of  values 
thus  arranged  according  to  magnitude  is  known  as  an 
array. 

An  inspection  of  the  five  cent  frequencies  shows  that 
we  have  discovered  only  a  very  rough  approximation 


12   INTEODUCTION  TO  ECONOMIC  STATISTICS 


TABLE  II 
WAGES  IN  A  CONNECTICUT  WOOLEN  MILL,  JULY,  1891 


Freqdency 

Frequency 

Daily 

Tally 

Classes— Inter- 

Daily 

Tally 

Classes — Inter- 

Wage 

(No.  of 

vals  of: 

2 

Wage 

(No.  of 

vals  OK : 

2 

$ 

Workers) 

$ 

Workers) 

5(J 

15<^ 

25^ 

5Q<i 

5(* 

15^ 

2.'-,(' 

50<^ 

.40 

Carried  over  . 

349 

351 

349 

349 

.45 

.50 

6- 

6 

5 

5 

2.50 

1-1- 

2 

351 

.55 

5 

2.55 

.60 

5 

5 

2.60 

0 

2 

.65 

0 

5 

2.65 

.70 

42 

5 

2.70 

6 

.75 

1-9- 

10 

15 

2.75 

1-1-1-1- 

4 

4 

355 

.80 

2-14- 

16 

29 

31 

2.80 

.85 

1-2- 

3 

37 

34 

2.85 

4 

.90 

1-3-4- 

8 

42 

2.90 

0 

.95 

10 



42 

2.95 

1.00 

1-1- 

2 

44 

3.00 

3- 

3 

358 

1.05 

2- 

2 

46 

3.05 

3 

1.10 

14-1-4- 

19 

38 

58 

65 

3.10 

3 

1.15 

1-1-3-12- 

17 

117 

82 

3.16 

1.20 

1-17- 

18 



100 

3.20 

0 

3 

1.25 

♦■1-1.3-3-2-4-3- 

28 

56 

128 

3.25 

858 

1.30 
1.85 

7-1-2- 
8-1-8-2- 

10 
19 

59 

13^ 
157 

3.30 
3.35 

0 

0 

1.40 

1-1- 

2 

21 

159 

3.40 

1.45 

159 

3.45 

1.50 

2-3-1-1-4- 

11 

170 

3.50 

1- 

1 

1 

359 

1.55 

14 

170 

3.55 

1.60 

2-1- 

3 

106 

173 

3.60 

1 

1.85 

1- 

1 

174 

8.65 

0 

1.70 

2-89- 

91 

169 

180 

265 

3.70 

1 

1.75 

1-3-2-60-1- 

67 

332 

3.75 

359 

1.80 

332 

3.80 

0 

1.85 

1-1- 

2 

7 

74 

334 

3.85 

0 

1.90 

1-4- 

5 

339 

3.90 

1.95 

339 

3.95 

1 

2.00 

1-1- 

2 

2 



341 

4.00 

1- 

1 

360 

2.05 

341 

4.05 

2.10 

2-1- 

3 

9 

844 

4.10 

0 

1 

2.16 

7 

344 

4.15 

2.20 

4- 

4 

10 

348 

4.20 

2 

2.J5 

1- 

1 

349 

4.25 

1- 

1 

1 

361 

2.30 

1 

349 

2.S5 

1 

349 

1 

2.40 

349 

2.46 

2 

349 

Totals  

349 

351 

349 

349 

Totals  

361 

361 

361 

361 

to  the  theoretical  frequency  curve.  We  therefore  ex- 
periment with  larger  groupings  to  see  if  we  can  thus 
obtain  more  distinctive  results.  Classes  at  fifteen, 
twenty-five,  and  fifty  cent  intervals  are  shown  in  the 
designated  columns.  These  classes  are  found  by 
adding  the  five  cent  frequencies  falling  within  the 
limits  indicated  by  the  horizontal  bars.     Obviously, 


TABULATION 


13 


several  variations  of  these  groupings  could  be  made 
by  beginning  at  different  points  in  the  scale;  but  the 
arrangement  here  chosen,  which  gives  regular  classes 
back  to  the  zero  point,  is  the  most  natural  one  to  take. 
Comparing  the  different  groupings,  we  see  that  the 
twenty-five  and  fifty  cent  intervals  give  results  as 
smooth  as  we  are  likely  to  get.  Since  a  classification 
with  larger  intervals  promises  to  be  too  indefinite,  it 
is  useless  to  carry  our  frequency  classifications 
further. 

The  Derived  or  Special  Purpose  Table.  In  order  to 
give  a  summarized  presentation  of  the  fifty  cent  fre- 
quencies. Table  III  has  been  drawn  up.  In  this  table 
the  classes  are  designated  by  stating  the  upper  and 
lower  limits,  as  $.50  to  $.95,  inclusive.  If  the  class 
interval  is  so  arranged  that  the  mid-point  falls  at  a 
round  number,  the  class  may  be  designated  by  this 
number.    Thus  the  twenty-five  cent  frequencies  could 

TABLE   III 
WAGES  IN  A  CONNECTICUT  WOOLEN  MILL  * 


NUMBER 

AND   PERCENTAGE 

OF  WORKERS  DISTRIBUTED 

ACCORDING  TO  DAILY  WAGES,  JULY 

WAGE  PER  DAY 

DOLLARS 

18 

70 

1880 

1891 

NO. 

% 

NO. 

% 

NO. 

% 

0    to       .45 

2 

1.8 

2 

0.9 

0 

0 

.50    to      .95 

18 

16.7 

58 

27.2 

42 

11.6 

1.00    to    1.45 

54 

50.0 

126 

59.2 

117 

32.4 

1.50    to    1.95 

22 

20.4 

17 

8.0 

180 

49.8 

2.00    to    2.45 

3 

2.8 

1 

0.5 

10 

2.8 

2.50    to    2.95 

6 

5.6 

3 

1.4 

6 

1.7 

3.00    to    3.45 

1 

0.9 

6 

2.8 

3 

0.8 

3.50    to    3.95 

2 

1.8 

0 

0 

1 

0.3 

4.00    to    4.45 

0 

0 

0 

0 

2 

0.6 

Total 

108 

100 

213 

100 

361 

100 

•  Aldrich  Report,  pp.  1463  ff. 


14  INTRODUCTION  TO  ECONOMIC  STATISTICS 

be  tabulated  according  to  the  nearest  quarter  dollar. 
The  usual  method  is,  however,  to  state  the  limits,  as 
here  shown. 

In  contrast  with  Table  I,  which  presented  in  orderly 
form  practically  all  the  detailed  data  of  our  study,  this 
table  is  a  derived  or  special  purpose  table.  It  aims  to 
present  only  certain  features  of  the  wage  schedules, 
and  therefore  purposely  omits  details.  It  is  in  the 
nature  of  a  generalization,  condensing  the  original 
facts  into  as  brief  a  compass  as  is  practicable. 

The  preparation  of  such  a  table  usually  calls  for 
both  consideration  and  skill.  The  bracketing  system 
used  in  the  headings,  or  captions,  is  obvious — the  wider 
blocks  bracket  and  designate  the  smaller  ones  imme- 
diately beneath.  Which  set  of  subdivisions  are  entered 
as  captions,  and  which  are  entered  in  the  stub  to  the 
left,  is  usually  determined  by  the  exigencies  of  space. 
The  arrangement  of  the  details  will  depend  upon  the 
nature  of  the  table.  In  census  tables,  for  example, 
the  current  date  is  placed  in  the  first  column,  to  the 
left,  because  of  its  greater  importance,  thus  reversing 
the  chronological  order.  In  such  tables,  also,  totals 
will  be  given  the  place  of  prominence  at  the  top, 
directly  beneath  the  caption.  In  an  elaborate  table 
percentage  columns  will  be  placed  together,  or  in  a 
separate  table,  to  allow  of  easy  comparison.  Correla- 
tive items  in  the  caption  or  stub  should  be  arranged 
in  some  logical  order,  whether  by  magnitude  as  in  the 
case  of  the  frequency  classes,  chronologically  as  the 
successive  wage  distributions,  geographically  as  in  the 
case  of  a  census  list  of  states,  by  order  of  origin,  or 
merely  alphabetically. 


TABULATION 


15 


In  computing  the  percentage  columns,  each  fre- 
quency is  divided  by  the  total.  The  work  will  be  suf- 
ficiently accurate  if  computed  on  a  slide  rule  or  string 
chart.  The  percentage  totals  will  not  necessarily  come 
to  exactly  one  hundred  per  cent,  because  of  the  inac- 
curacies involved  in  cutting  off  decimals.  If  it  is  de- 
sired, however,  they  may  be  brought  to  the  proper 


60. 


/S70    — 

/S3  0 

/S9/    = 


/  m  2.00  doo 

Figure  2.     Frequency  polygons 

total  merely  by  turning  one  or  two  items  that  stand 
at  or  near  five  in  the  first  decimal  dropped.  Thus 
7.55  may  be  written  7.6  or  7.5,  according  to  which  is 
needed  to  make  up  the  total  of  one  hundred;  or  7.56 
might  even  be  written  7.5  if  no  better  can  be  done.  A 
sufficient  degree  of  accuracy  can  always  be  secured  by 
extending  the  number  of  decimals  retained.  But,  in 
general,  the  special  purpose  table  should  not  exhibit 
meticulous  accuracy.     Decimals  should  be  shortened 


16  INTRODUCTION  TO  ECONOMIC  STATISTICS 

or  dropped,  fractions  avoided,  and  large  numbers 
rounded. 

The  Frequency  Polygon.  The  percentage  columns 
of  Table  III  may  now  be  graphically  presented  by 
''frequency  polygons,"  as  shown  in  Figure  2.  The 
percentage  columns  are  here  used  in  preference  to 
the  absolute  numbers  because  they  reduce  the  three 
polygons  to  the  same  scale.  It  will  be  seen  that  in 
drawing  the  frequency  polygons  the  points  represent- 
ing the  percentage  frequencies  are  plotted  directly 
above  the  mid-point  of  each  class,  respectively.  These 
points  are  then  joined,  the  individual  years  being  dis- 
tinguished by  different  kinds  of  lines.  The  graph 
brings  out  very  well  the  general  advance  in  minimum, 
maximum,  and  mean  Avages  that  occurred  in  1891  in 
the  mill  under  consideration.  The  data  for  a  single 
year  may  also  be  graphed  as  a  ''rectangular  histo- 
gram," as  illustrated  in  the  next  chapter  (Fig.  3,  p.  25; 
solid  line).  Both  the  frequency  polygon  and  the  rec- 
tangular histogram  are  frequency  curves  approxi- 
mately expressed. 

Difficult  Features  of  Tabulation.  Before  leaving  the 
subject  of  tabulation,  a  few  general  suggestions  may 
be  made.  In  the  tabulation  considered  in  this  chapter, 
the  problem  of  interpreting  the  term  "wages"  has 
been  solved  for  us  by  the  Aldrich  Report.  But  if  we 
had  undertaken  the  task  of  filling  in  the  original 
schedules  by  actual  field  work,  we  should  have  been 
faced  with  the  difficulty  of  drawing  a  somewhat  arti- 
rficial  line  distinguishing  wages  from  salaries,  and 
perhaps  from  commissions  and  other  direct  or  indirect 
income.  From  the  standpoint  of  economic  theory,  of 
course,  salaries  are  generically  wages.    But  in  practice 


TABULATION  17 

payments  for  relatively  responsible  and  skilled  work, 
contracted  usually  on  the  basis  of  a  considerable  period 
of  time,  and  carrying  some  degree  of  stability  of 
tenure,  are  classed  as  salaries.  They  are  excluded 
from  wage  schedules  as  being  presumably  determined 
less  directly  by  supply  and  demand  considerations. 
Likewise  most  statistical  units,  however  precise  and 
simple  they  may  appear  at  first  glance,  usually  present 
many  difficulties  when  they  are  applied  to  real  condi- 
tions. Precisely  what,  for  example,  should  be  included 
in  a  tabulation  as  a  book,  a  farm,  an  accident,  a  ton- 
mile?  Almost  any  unit  that  may  be  chosen  will  be 
found  to  call  for  careful  discrimination,  and  an  exam- 
ination of  current  usage. 

A  further  difficulty  is  encountered  when  data  con- 
cerning given  units  are  being  gathered  and  compared 
over  a  certain  period  of  time,  or  from  different  con- 
temporaneous environments.  It  often  happens  that 
the  definition  of  the  unit  varies  at  different  times  or 
in  different  places ;  or  perhaps  the  basis  of  estimating 
the  frequencies  may  be  altered,  or  comparisons  may 
be  invalidated  by  changing  conditions.  In  our  study  of 
the  Connecticut  mill  we  may  evidently  assume  that 
the  basis  of  the  classification  has  not  changed  materi- 
ally through  the  period  studied,  since  the  occupations 
classed  as  wage-earning  are  specified.  BT;t  we  might 
modify  our  interpretation  of  the  change  in  wage  levels 
upon  observing  that  processes  of  work  had  altered, 
that  the  work-day  was  shortening,  that  child  labor 
legislation  was  affecting  the  personnel,  that  the  per- 
centage of  female  workers  shifted  from  28%  to  19%, 
and  then  to  32%,  or  that  the  cost  of  living  had  fallen. 
Thus  it  is  always  necessary  in  comparative  studies  to 


18  INTRODUCTION  TO  ECONOMIC  STATISTICS 

consider  carefully  both  the  environmental  conditions 
and  the  statistical  units  employed. 

The  statistician  who  is  at  all  ingenious  will  discover 
many  short  cuts  to  lighten  the  work  of  tabulation.  A 
method  which  is  often  useful  is  that  of  entering  the 
original  data  on  3  X  5  or  4  X  6  cards.  Suppose,  for 
example,  that  we  wished  to  classify  the  students  of  a 
given  college  according  to  their  entrance  grades,  the 
class  of  school  from  which  entering,  fraternity  mem- 
bership, and  scholastic  standing  during  their  college 
course.  A  numbered  card  could  be  prepared  for  each 
student,  and  the  desired  data  entered.  The  cards  could 
then  be  sorted  as  desired,  the  sub-totals  could  be  de- 
termined, or  the  data  listed.  If,  however,  such  work 
is  to  be  done  on  an  extended  scale,  a  tabulating  ma- 
chine will  be  required.  Such  a  machine  automatically 
sorts  and  counts  special  cards  on  which  the  required 
data  have  been  recorded  by  a  keyed  punch.  The  larger 
business  houses  are  using  machine  tabulators  increas- 
ingly; and  of  course  extensive  compilations  like  a  cen- 
sus are  prepared  principally  by  machines. 

Library  Work.  The  subject  of  tabulation  has  been 
extensively  treated  by  writers  on  statistics.  Day's 
article,  cited  below,  will  prove  to  be  very  valuable  to 
the  student.  It  may  be  found  reprinted  in  Secrist's 
** Readings,"  together  with  another  excellent  article 
on  the  same  subject.  Chapter  IV  of  the  same  book  is 
especially  pertinent  to  the  subject  of  wage  tabulations. 
Chapter  VII  of  Rugg's  text-book  presents  a  concise 
description  of  the  frequency  curve.  Machine  tabula- 
tion is  described  in  the  circulars  of  the  Tabulating 
Machine  Company,  of  New  York. 


TABULATION  19 

REFERENCES 

Bailey  and  Cummings,  Statistics,  Chapters  I-V. 

Bowley,  Arthur  L.,  Elements  of  Statistics,  Chapter  IV. 

Day,  E.  E.,  "Standardization  of  the  Construction  of  Statis- 
tical Tables,"  Quarterly  Publications  pf  the  American 
Statistical  Association,  March,  1920,  pp.  59-66. 

Koren,  John,  A  History  of  Statistics. 

Rugg,  H.  0.,  Statistical  Methods  Applied  to  Education,  Chap- 
ter VII. 

Secrist,  Horace,  An  Introduction  to  Statistical  Methods, 
Chapters  I-V. 

Secrist,  Horace,  Readings  and  Problems  in  Statistical  Meth- 
ods, Chapters  I-V. 

Yule,  G.  U.,  An  Introduction  to  the  Theory  of  Statistics, 
Chapter  VI. 

EXERCISES  1 

1.  Draw  up  tally  sheets  and  frequency  classifications  for  the 
years  1870  and  1880. 

2.  Tabulate  the  twenty-five  cent  classes,  showing  both  abso- 
lute and  percentage  figures,  for  the  years  1870,  1880,  and 
1891.  Draw  frequency  polygons  from  the  percentage 
data. 

3.  Similarly  tabulate  and  graph  the  fifteen  cent  classes. 

4.  Graph  the  five  cent  classes  for  the  years  1870,  1880,  and 
1891. 

5.  Draw  rectangular  histograms  of  the  fifty  cent  frequencies 
for  1870  and  1880.  Compare  the  relative  advantages  of 
the  frequency  polygon  and  the  rectangular  histogram. 

6.  Classify  and  tabulate  separately  the  female  workers  for 
the  years  1870,  1880,  and  1891.  (Starred  items— see  foot- 
note. Table  I.) 

7.  Obtaining  data  from  the  Aldrich  Report,  study  the  wages 
paid  in  Establishment  No.  86  in  July  of  1875  and  1885. 
Classify,  tabulate,  and  graph  as  for  the  other  years 
studied. 

8.  Obtain  the  average  scholarship  grades  for  a  selected  group 
of  students  (100  or  more),  classify  these  grades,  tabulate, 
and  draw  a  frequency  polygon. 

9.  Toss  two  coins  twenty-five  times,  keeping  a  record  of  the 
number  of  heads  thrown  at  each  toss.  Classify  and  tabu- 
late the  results,  and  draw  a  frequency  polygon.  Toss 
four  coins  fifty  times,  making  similar  records.  What  prin- 
ciple is  illustrated? 

*  Before  beginning  a  notebook  the  student  abould  read  Appendix  I, 


CHAPTER  II 

TYPES  AND  MEASURES  OF  DISPERSION 

After  the  frequency  distribution  of  a  given  array 
has  been  presented  in  suitable  form,  there  remains  the 
task  of  finding  simple  numerical  measures  by  which 
it  may  be  summed  up  for  purposes  of  ready  descrip- 
tion and  comparison.  The  two  features  thus  to  be 
expressed  are  the  typical  wage  and  the  degree  of  dis- 
persion or  ** spread"  about  the  type.  As  a  wage  type, 
and  a  base  from  which  dispersion  may  be  measured, 
the  common  average,  or  arithmetic  mean,  will  immedi- 
ately suggest  itself.  In  the  case  of  the  wage  rolls  given 
in  the  preceding  chapter,  the  average  may  be  most  con- 
veniently found  by  multiplying  the  wages,  as  tabulated 
at  five  cent  intervals,  by  their  respective  frequencies. 
The  sum  of  these  products,  divided  by  the  number  of 
workers,  is  the  average.  It  is  the  weighted  average  of 
the  class  values  at  five  cent  intervals,  since  these  values 
are  emphasized  in  accordance  with  their  frequencies. 
We  shall  find  that  weighted  averages  are  sometimes 
taken  in  which  the  weights  are  derived  estimates  of 
the  importance  which  should  be  attached  to  the  values, 
respectively;  but  in  this  case  the  weighting  amounts 
simply  to  a  summing  up  of  the  original  wages.    The 

formula  for  the  weighted  average  is  — ^rj—    (summa- 
tion of  the  frequencies  times  the  class  values,  divided 

20 


TYPES  AND  MEASUEES  OF  DISPERSION      21 

by  the  number  of  items).  The  accompanying  table 
(Table  IV),  derived  from  Table  II,  shows  the  process 
of  finding  the  average  wage  for  the  year  1891. 

TABLE  IV 
WAGE  EOLL  AND  AVERAGE  WAGE 

CONNECTICUT    MILL,    JULY,    1891 


Single  Wage 

No.  of  Workers 

Total  Wage 

$  .50 

5 

$2.50 

.75 

10 

7.50 

.80 

16 

12.80 

.85 

3 

2.55 

.90 

8 

7.20 

1.00 

2 

2.00 

1.05 

2 

2.10 

1.10 

19 

20.90 

1.15 

17 

19.55 

1.20 

18 

21.60 

1.25 

28 

35.00 

1.30 

10 

13.00 

1.35 

19 

25.65 

1.40 

2 

2.80 

1.50 

11 

16.50 

1.60 

3 

4.80 

1.65 

1 

1.65 

1.70 

91 

154.70 

1.75 

67 

117.25 

1.85 

2 

3.70 

1.90 

5 

9.50 

2.00 

2 

4.00 

2.10 

3 

6.30 

2.20 

4 

8.80 

2.25 

1 

2.25 

2.50 

2 

5.00 

2.75 

4 

11.00 

3.00 

3 

9.00 

3.50 

1 

3.50 

4.00 

1 

4.00 

4.25 

1 

4.25 

Total 

361 

$541.35 

1.49958 

Average 

$1.50 

The  Mode.    In  addition  to  the  arithmetic  mean,  there 
are  other  types  which  the  statistician  uses  in  summar- 


22  INTEODUCTION  TO  ECONOMIC  STATISTICS 

izing  an  array  and  measuring  dispersion.  One  of  these 
is  the  mode.  The  mode  is  applicable,  however,  only 
to  frequency  distributions  which  conform  in  their  gen- 
eral outlines  to  the  theoretical  frequency  curve.  It  is 
the  value  which  lies  at  the  point  of  greatest  frequency, 
and  in  the  normal  curve  is  therefore  identical  with 
the  average.  In  the  graph  of  a  frequency  distribution 
it  is  easily  recognizable  as  the  value  indicated  on  the 
horizontal  scale  at  the  point  directly  under  the  highest 
point  of  the  curve.    (See  Figure  3,  page  25.) 

The  mode  is  particularly  useful  in  connection  with 
those  frequency  curves  which,  though  conforming  in 
general  outlines  to  the  theoretical,  are  extended  more 
on  the  one  side  than  the  other.  Such  curves  are  said  to 
be  skewed.  The  wage  data  studied  in  the  preceding 
chapter  gives  curves  which  are  somewhat  skewed  to 
the  right,  so  that  a  small  secondary  mode  sometimes 
appears.  But  in  their  original  five  cent  frequencies 
they  do  not  give  a  smooth  enough  curve  to  allow  of 
a  very  definite  mode.  When,  however,  such  curves  are 
strongly  skewed,  and  are  yet  passably  smooth,  the 
mode  is  preferable  to  the  average  as  a  type  of  the 
array.^  Suppose,  for  example,  that  in  a  wage  array  a 
few  very  large  salaries  are  included.  In  such  a  case 
the  average  may  fall  between  the  wages  and  the 
salaries,  at  a  point  where  the  frequencies  are  small. 
The  mode,  on  the  other  hand,  states  the  wage  or  salary 
most  frequently  paid.  It  is  not  affected  by  the  skewed 
extreme  of  the  curve;  that  is,  by  the  relatively  small 
number  of  large  salaries. 

'  Frequency  curves  that  are  strongly  skewed  to  the  right  will  some- 
times    appear    normal    if    transferred    to    semi-logarithmic    paper,    the 


TYPES  AND  MEASURES  OF  DISPERSION      23 

Determining  the  Mode.  In  frequency  distributions 
that  are  irregular  the  mode  may  often  be  approxi- 
mately determined  by  a  study  of  the  larger  frequency 
groupings.  Let  us  take  as  an  illustration  the  twenty- 
five  cent  frequency  classes  derived  from  the  1891  wage 
data.  These  are  shown  in  the  third  column  of  Table  V, 
the  first  and  second  columns  being  the  data  from  which 
they  are  derived,  as  given  in  Table  II.  The  latter  part 
of  the  series  is  omitted,  however,  since  it  cannot  affect 
the  position  of  the  mode.  In  the  fourth  and  succeed- 
ing columns,  variations  of  the  twenty-five  cent  fre- 
quencies are  formed  by  beginning  the  classes  at  differ- 
ent points  in  the  scale.  In  each  case  the  mode  should 
lie  somewhere  within  the  class  having  the  largest 
frequency ;  that  is,  it  should  lie  in  the  following  classes : 
$1.50  —  $1.70 

1.55  —    1.75 

1.60  —    1.80 

1.65  —    1.85 

1.70  —  1.90 
Since  the  only  point  common  to  these  five  classes  is 
$1.70,  this  sum  may  be  regarded  as  the  mode.  It  will 
be  seen,  however,  that  this  is  the  same  value  which 
would  be  taken  as  the  mode  on  the  basis  of  the  five  cent 
classes.  The  same  result  would  in  this  case  also  be 
obtained  by  using  the  fifty  cent  classes.  Ordinarily, 
this  method  is  applied  only  to  the  largest  classes  which 
it  is  practicable  to  use  in  the  frequency  classification. 
It  may  give  a  quite  different  result  from  that  which 
is  obtained  from  the  smallest  classes. 

logarithmic  scale  being  used  as  the  base  line.  When  this  is  the  case, 
the  distribution  is  normal  on  the  basis  of  the  geometric  rather  than 
the  arithmetic  mean  (see  page  94). 


24  INTEODUCTION  TO  ECONOMIC  STATISTICS 


TABLE  V 

THE  MODE 

WAGE  EOLL,  CONNECTICUT  MILL,  JULY,  1891 


V 

F 

FREQUENCIES  IN  25c  CLASSES 

VARIOUS  GROUPINGS 

SUMMARY 

$  .40 

0 

5 

5 

.45 

0 

5 

5 

.50 

5 

5 

5 

.55 

0 

5 

5 

.60 

0 

5 

5 

.65 

0 

10 

10 

.70 

0 

26 

26 

.75 

10 

29 

29 

.80 

16 

37 

37 

.85 

3 

37 

37 

.90 

8 

29 

29 

.95 

0 

15 

15 

1.00 

2 

31 

31 

1.05 

2 

40 

40 

1.10 

19 

58 

58 

1.15 

17 

84 

84 

1.20 

18 

92 

92 

1.25 

28 

92 

92 

1.30 

10 

77 

77 

1.35 

19 

59 

59 

1.40 

2 

42 

42 

1.45 

0 

32 

32 

1.50 

11 

16 

16 

1.55 

0 

15 

15 

1.60 

3 

106M 

106 

1.65 

1 

162M 

162 

1.70 

91 

162M 

162M 

1.75 

67 

161M 

161 

1.80 

0 

165M 

165 

1.85 

2 

74 

' 

74 

1.90 

5 

9 

9 

1.95 

0 

9 

9 

2.00 

2 

10 

10 

2.05 

0 

5 

5 

2.10 

3 

9 

9 

2.15 

0 

8 

8 

2.20 

4 

8 

8 

2.25 

1 

5 

5 

2.30 

0 

5 

5 

2.35 

0 

1 

1 

2.40 

0 

2.45 

0 

etc. 

TYPES  AND  MEASURES  OF  DISPERSION      25 

As  applied  to  the  given  wages,  however,  the  fore- 
going method  of  locating  the  mode  is  objectionable. 
The  $1.70  frequency  is  relatively  so  large  that  in  each 
classification  it  determines  the  mode  without  allowing 
due  weight  to  the  large  frequencies  a  little  further 
down  the  scale.  The  latter  might  be  considered  a  sec- 
ondary mode,  but  we  shall  here  assume  that  the  use  of 
more  extensive  data  would  result  in  a  single  mode. 
We  may  therefore  illustrate  the  use  of  a  method  which 


200A 


I 

\ 


/ 
/ 
/ 

J— 


tVa^e: 


JO 


1.00 


—I — 
I.SO 


H»l  I    ■%   M  ,1 

— I — 
Z.00 


2.J0 


jno       J-^O        4-00       4.S0 


Figure  3.  Eectangiilar  histogram  of  wages  in  a  Connecticut  mill, 
1891,  fifty  cent  classes  (solid  line),  and  smoothed  frequencies  (broken 
line). 

is  applicable  to  irregular  frequencies,  or  to  data  pre- 
sented in  only  a  few  large  classes.  It  will,  of  course, 
be  understood  that  with  such  limited  data  no  very  de- 
pendable result  can  be  obtained. 

In  using  this  method,  we  shall  not  only  approxi- 
mately determine  the  mode,  but  draw  the  smoothed 
curve  as  well.  The  method  is  illustrated  in  Figure  3. 
The  frequencies  are  first  represented  by  a  rectan- 
gular histogram.  In  drawing  the  histogram  the  ver- 
tical lines  should  theoretically  be  drawn  at  a  point 


26  INTRODUCTION  TO  ECONOMIC  STATISTICS 

midway  between  the  two  adjacent  class  limits;  for 
example,  the  line  separating  the  first  and  second 
classes  falls  at  .475.^  It  is  evident  that  the  mode  lies 
in  the  class  $1.50  to  $1.95,  but  it  is  desirable  to  locate 
it  somewhat  precisely  within  the  class.  This  may  be 
done  by  dividing  the  class  interval  into  two  parts  pro- 
portional inversely  to  the  adjacent  frequencies.^  In  so 
doing  the  class  limits  are  considered  $1,475  and  $1,975, 
as  drawn.  The  division  may  readily  be  constructed 
geometrically,  or  it  may  be  computed  as  follows ; 
10 

$1,475  H X  $.50  =  $1.51 

117  +  10 

The  formula  for  this  operation  is, 

•pi 

M  -  Li  + ^ X  C 

Fm+Fn 

in  which 

M  =  the  mode. 

Lj  =  the  lower  limit  of  the  modal  class. 

Fm  and  Fn  =  frequencies  adjacent  to  the  one  contain- 
ing the  mode,  in  the  order  named. 
C  =  the  class  interval. 
Smoothing  the  Frequencies.    After  the  mode  has 
been  determined,  the  histogram  may  be  smoothed  into 
a  frequency  curve.     This  curve  is  drawn  to  conform 
as  closely  as  possible  to  the  theoretical  bell-shaped 
curve;  and  yet  to  maintain,  frequency  by  frequency, 
the  same  area  as  the  original  rectangular  figure.  Thus 
in  the  drawing  A^  =  Aj  +  ^3?  ^i  =  ^2,  and  Ci  =  Cj. 
The  curve  culminates  approximately  at  the  mode  as 

*  The  clasa  limits  should  be  so  placed  that  the  items  in  the  modal  class 
average  close  to  the  mid-point  of  the  class. 

'  It  is  sometimes  iireferable  to  take  the  average  of  the  items  in  the 
modal  and  adjacent  classes,  or  to  include  more  classes. 


TYPES  AND  MEASURES  OF  DISPERSION      27 

previously  determined,  but  the  height  is  merely  esti- 
mated with  reference  to  the  required  area.  As  thus 
drawn,  the  curve  presents  an  estimate  of  the  probable 
distribution  of  the  economic  values  expressed  by  wage 
rolls  of  the  type  studied.  The  irregularity  of  the  data, 
however,  makes  it  far  from  typical. 

The  Median.  Another  type  often  used  to  represent 
a  given  array  is  the  median.  The  median  is  the  value  of 
the  middle  item  in  an  array.    The  number  of  this  item 

N  +  l^ 
is  found  by  the  formula, ;  and  its  value  may  be 

2 

determined  by  reference  to  the  summation  column  of 
the  frequency  table.  For  example,  in  the  1891  wage 
data  the  median  item  is  number  181,  and  its  value  as 
determined  by  means  of  the  summation  column  is 
$1.70.  That  is,  the  181st  item  falls  within  the  $1.70 
class.  In  case  the  median  item  should  prove  to  be 
fractional,  and  should  fall  between  two  frequencies,  the 
median  would  not  be  precisely  determined.  Suppose, 
for  example,  that  the  median  item  had  been  number 
174^.  In  such  a  case  the  median  would  lie  between 
the  limits  $1.65  and  $1.70. 

Comparison  of  the  Three  Types.  Theoretically,  in  a 
frequency  curve  having  very  small  class  intervals  the 
median  value  is  indicated  at  the  foot  of  a  perpendic- 
ular line  which  bisects  the  area  of  the  curve.  The  aver- 
age, on  the  other  hand,  would  lie  at  the  foot  of  a  similar 
perpendicular  which  would  balance  as  an  axis  the 
weight  of  the  two  sides,  supposing  the  area  of  the  curve 

*  The  unit  is  added  to  counterbalance  the  space  from  0  to  1.  Or,  the 
formula  may  be  considered  as  an  expression  of  the  average  of  the 
extreme  ordinals  of  the  array. 


28  INTRODUCTION  TO  ECONOMIC  STATISTICS 

to  have  been  cut  out  of  a  material  of  uniform  weight. 
When  skewness  is  regular,  the  mode,  median,  and 
average  are  located  on  the  value  scale  in  the  order 
named,  the  intervals  separating  them  being  in  the  ratio 
of  about  two  to  one.  In  the  normal  curve  the  three 
types  are  identical.^ 

The  use  of  the  average,  median,  and  mode  as  types 

SALARIES    PAID    IN    REPRESENTATIVE    UNIVERSITIES    AND 

COLLEGES  IN   THE  UNITED  STATES   IN   1919-20. 

Public  institutioTis 


TITLE  OF  POSITION 


President  or  chancellor. 

Dean  or  director , 

Professor 

Associate  professor.  . .  .  , 
Assistant  professor.  .  .  . 

Instructor 

Assistant , 


NUMBER  OP 
PERSONS 


77 

367 

2,460 

822 

1,705 

2,138 

855 


MINIMUM 
SALARY 


$2,500 
1,200 

300 
300 
500 
300 
75 


MAXIMUM 
SALARY 


$12,500 

10,000 
10,000 
4,000 
4,000 
3,100 
2,500 


President  or  chancellor. 

Dean  or  director , 

Professor 

Associate  professor.  . . . 
Assistant  professor.  . . . 

Instructor 

Assistant , 


AVERAGE 
SALARY 


$6,647 
3,819 
3,126 
2,514 
2,053 
1,552 
801 


MEDIAN 
SALARY 


$6,000 
3,500 

3,000 
2,500 
2,000 
1,500 
750 


MOST 

FREQUENT 

SALARY 


$6,000 

3,000 
3,000 
3,000 
1,800 
1,500 
1,200 


*  Two  other  forms  of  the  average  are  sometimes  used  in  statistical 
work.  One  is  the  geometric  mean.  This  may  be  found  by  averag- 
ing the  logarithms  of  the  numbers  instead  of  the  numbers  themselves. 
It  ia  the  Jith  root  of  the  product  of  the  numbers.  Some  statisticians 
advocate  the  use  of  the  geometric  mean  in  finding  the  average  periodic 
change  in  prices.  Considered  merely  as  an  average  of  prices  apart  from 
the  use  of  weights,  the  geometric  mean  is  logically  correct  because  it 
measures  ratios  of  divergence  rather  than  absolute  amounts.  The  other 
type  of  average  is  the  harmonic  mean,  which  has  occasionally  been 
applied  to  the  same  purpose.    For  two  numbers,  a  and  b,  it  is  computed 

2  ab 
by  the  formula  — —r--     In  arithmetic  this  is  the  formula  which  is  used  to 
a  -f  b 

find  an  average  rate  of  travel  when  two  rates  for  two  equal  distances  are 

given.     In  general,  it  may  be  described  as  the  reciprocal  of  the  average 

of  the  reciprocals  of  the  given  numbers. 


TYPES  AND  MEASURES  OF  DISPERSION      29 

may  be  illustrated  by  the  foregoing  table  which  was 
issued  by  the  United  States  Bureau  of  Education  and 
reprinted  in  the  Monthly  Labor  Review  of  January, 
1921.  In  this  table  the  term  ''most  frequent  salary" 
signifies  the  mode. 

Quartile  Deviation.  The  types  we  have  now  consid- 
ered are  used  as  the  basis  for  measuring  dispersion, 
though  the  average  is  more  commonly  employed  than 
the  other  two.  The  simplest  measure  of  dispersion  is 
related  to  the  median,  and  is  called  the  quartile  devia- 
tion. This  is  found  by  computing  the  value  of  the  first 
and  third  quartiles  of  an  array.  The  quartiles  are 
analogous  to  and  include  the  median,  being  located  at 

the  quarter  divisions  of  the  array.    The  location  of  the 

I   -J 

first  quartile  is  found  by  the  formula   — j — -,  and  of 

the  third  quartile  by  the  formula j — .  The  values 

of  these  items  are  determined  by  reference  to  the  fre- 
quency table  in  the  same  manner  as  the  median  value 
was  determined.  The  second  quartile  is,  of  course, 
identical  with  the  median.  The  quartile  range  is  found 
by  subtracting  the  value  of  the  first  quartile  from  the 
value  of  the  third,  and  the  quartile  deviation  is  half 
of  this  difference.  It  may  be  seen  by  reference  to 
Figure  1  that  the  quartile  deviation  as  thus  found  is 
simply  the  average  distance  between  the  median  and 
the  adjacent  quartiles,  as  measured  on  the  base  line. 
In  the  1891  wage  roll  the  first  quartile  is  item  No.  90yo, 
and  its  value  is  $1.20.  The  third  quartile  is  No.  271i/2» 
and  its  value  is  $1.75.  The  quartile  range  is  therefore 
$.55,  and  the  quartile  deviation  is  $.28.     This  means 


30  INTRODUCTION  TO  ECONOMIC  STATISTICS 

that  half  the  workers  receive  wages  that  fall  within  a 
range  averaging  twenty-eight  cents  above  and  below 
the  median ;  that  is,  between  $1.20  and  $1.75.^ 

For  purposes  of  comparison  the  quartile  deviation 
should  usually  be  reduced  to  a  percentage  basis.  This 
is  done  by  dividing  it  by  a  value  regarded  as  typical  of 
the  array.  Since  the  quartile  deviation  is  related  to 
the  median,  it  would  appear  logical  to  take  this  type 
as  a  base.  But  it  is  customary  to  take  instead  a  point 
lying  midway  between  the  first  and  third  quartile 
values ;  that  is,  the  average  of  the  two.  In  a  perfectly 
regular  curve  this  value  would  naturally  be  identical 
with  the  median.  The  reason  for  taking  this  base  is 
obviously  that  it  is  the  point  from  which  the  quartile 
deviation  is  assumed  to  be  directly  measured.     The 

Q3+Q1 
formula  for  it  is  (third  quartile  plus  first 

2 

quartile,  divided  by  two).  The  formula  for  the  quartile 

Q3-Q1 
deviation  is  .      The  latter  divided  by  the 

2 

former  is ,  which  is  therefore  the  formula  for 

Q3+Q1 
the  coefficient  of  quartile  deviation. 

Interpolation.  Before  leaving  the  quartiles,  a  method 
of  locating  them  by  interpolation  between  the  class 
intervals  should  be  described.  We  will  illustrate  the 
method  by  applying  it  to  the  1891  wage  data,  though 

*  The  quartile  deviation  is  also  called  the  "probable  error"  of  a 
frequency  distribution.  The  term  is  derived  from  the  ' '  Theory  of 
Errors, ' '  and  connotes  the  central  range  of  the  distribution  within 
which  an  item  added  to  tho  series  will  have  an  even  chance  of  fall- 
ing. 


TYPES  AND  MEASUEES  OF  DISPERSION      31 

in  fact  the  five  cent  classes  give  quartile  values  precise 
enough  for  most  purposes.  The  type  of  problem  to 
which  the  method  is  best  adapted  is  one  in  which  an 
array  as  given  is  classified  only  in  a  few  large  groups. 
But  supposing  that  it  is  desirable  to  know  the  quartile 
values  very  precisely  in  the  1891  wage  data,  we  may 
find  them  as  described  below. 

In  assuming  that  we  may  interpolate  at  any  point 
between  the  items  of  the  original  frequency  classifica- 
tion, we  are  evidently  regarding  the  series  as  con- 
tinuous rather  than  discrete.  In  the  case  of  the  wage 
roll  we  shall  be  dealing,  then,  with  the  theoretical 
economic  values  underlying  the  wages  as  paid.  The 
actual  frequencies  are  therefore  to  be  considered  as 
indicating  proportionate  numbers,  which  may  be  in- 
creased indefinitely  as  in  the  case  of  any  multiple  ratio. 
The  original  discrete  five  cent  classes  are  now  to  be 
considered  as  having  continuous  class  intervals;  for 
example,  a  fifty  cent  wage  is  taken  to  indicate  an 
economic  value  within  the  limits  $.475  and  $.525. '  The 
frequencies  are  assumed  to  be  equally  spaced  between 
these  limits. 

When   about   to    interpolate,   we   locate   the   first 

N  N  +  1 

quartile  by  the  formula  — ,  instead  of as  in  the 

4  4 

previous  case.  The  second  and  third  quartiles  are 
two  and  three  times  this  number,  respectively.  The 
reason  for  omitting  the  unit  in  the  formula,  and  for 
ignoring  it  also  in  an  analogous  formula  for  sub-divid- 
ing the  class,  is  that  our  hypothesis  of  a  continuous 
scale  renders  the  unit  of  negligible  value.    It  is  as  if 


32  INTRODUCTION  TO  ECONOMIC  STATISTICS 

the  items  were  regarded  as  being  groups  of  thousands 
or  millions;  that  is,  as  if  they  were  indefinitely  sub- 
divisible. Having  located  the  quartile  items,  we  find 
the  class  in  which  they  fall  by  inspection  of  the  fre- 
quency table,  as  before.  We  next  find  the  position  of 
the  quartile  within  the  class;  that  is,  the  fraction  of 
the  interval  that  it  is  advanced  beyond  the  lower  limit. 
The  corresponding  value  is  then  determined.  The 
process  is  the  same  as  that  used  in  interpolating  in 
logarithmic  or  other  tables. 

In  the  1891  wage  data,  the  first  quartile  is  located  at 
item  901/4.  This  item  falls  in  the  class  having  theo- 
retically a  lower  limit  of  $1,175  and  an  upper  limit  of 
$1,225.  The  preceding  class  ends  with  the  82nd  item, 
and  the  quartile  is  therefore  advanced  8^/4  items  in  its 
own  class  of  18  items.  This  advance  is  8^1  -^  18,  or  .46 
of  the  class  interval.  This  fraction  of  the  class  interval 
of  $.05,  is  $.023,  which,  added  to  the  lower  limit  of  the 
class,  gives  the  quartile  value  of  $1,198.  Read  to  the 
nearest  cent,  this  value  happens  to  be  the  same  as  that 
obtained  without  interpolation. 

The  process  may  be  summed  up  as  follows : 
I 

Q  =  L,  +  — .  C 
F 
in  which, 

Q   =  Quartile  value 

Lj  =  Lower  limit  of  class  containing  quartile 

I     =  Quartile  item  minus  last  item  of  preceding 

class  ^ 
F   =  Frequency  of  class  containing  quartile 
C    =  Interval  of  same  class 

*"Item"  here  refers  to  the  number,  not  the  value. 


TYPES  AND  MEASURES  OF  DISPERSION      33 


Quartile  Dispersion.  A  statement  of  the  quartiles 
and  the  highest  and  lowest  wage  paid  serves  to  give  a 
fairly  good  idea  of  a  frequency  distribution  even  with- 
out any  further  computation  of  precise  measures  of 
dispersion.    In  Table  VI  such  a  statement  is  presented 

EANGE  AND  TREND  OF  WAGES,  1870-1891,  IN  A  CONNECTICUT 
WOOLEN  MILL 


4.oo~- 

J.OO. 
Z.00. 

/.so. 

1.00. 


.so. 


;5o. 


/i=/7j^/?esT;     Q,-G^=^uarr/fe5,   L  = /on^esT  wage 


H 


1370 


mo 


mo 


Figure  4.     Semi-logarithmic,  or  ratio  paper 

for  the  wage  data  of  1870,  1880,  and  1891,  together 
with  the  quartile  deviations  and  their  coefficients.  The 
table  includes  the  interpolated  values,  although,  as  has 
been  intimated,  their  computation  is  hardly  worth 
while  here  except  as  an  illustration  of  the  method.  In 
Figure  4  the  discrete  quartiles  and  limits  are  shown 


34  INTRODUCTION  TO  ECONOMIC  STATISTICS 

graphed  upon  semi-logarithmic  paper.  The  vertical 
scale  of  this  paper  is  similar  to  the  scale  of  a  slide- 
rule,  hence  the  ratio  of  periodic  change  may  be  com- 
pared by  means  of  the  slant  of  the  lines  connecting 
the  values.     To  be  complete,  however,  such  grauhic 

TABLE  VI 

RANGE  OF  DAILY  WAGES  IN  A  CONNECTICUT  MILL— SCALE 
TAKEN  BOTH  AS   DISCRETE   AND   CONTINUOUS 


WAGE 

1870 

1880 

1891 

ARRAY 

DIS- 
CRETE 

CONTINU- 
OUS 

DIS- 
CRETE 

CONTINU- 
OUS 

DIS- 
CRETE 

CONTINU- 
OUS 

Lower  Limit 

$    .40 
1.10 
1.30 

1.50 
3.75 

$    .38 

1.09 
1.30 
1.51 

3.78 

$    .40 

.95 
1.20 
1.25 
3.25 

$  .38 

.93 

1.19 

1.24 

3.28 

$    .50 
1.20 
1.70 
1.75 
4.25 

$    .48 

1st  Quartile 

2nd  Quartile 

1.20 
L68 

3rd  Quartile 

1.73 

Higher  Limit 

4.28 

Q.  Deviation 

.20 
15% 

.21 
16% 

.15 

14% 

.16 

14% 

.28 
19% 

.27 

— coefficient 

18% 

representation  should  show  at  least  annual  data.  It 
might  also  very  well  show  decile  points;  that  is,  the 
wages  occurring  at  the  tenths  instead  of  the  quarters 
of  each  array. 

The  Ogive.  A  convenient  method  of  presenting  an 
array  and  at  the  same  time  of  graphically  determining 
the  quartiles,  is  shown  in  Figure  5.  The  construction 
is  based  upon  the  assumption  of  a  continuous  series  of 
values,  and  parallels  the  procedure  of  finding  the  quar- 
tile values  by  interpolation.  The  frequencies  are 
plotted  from  the  summation  column  of  the  original 
five  cent  classes.  For  convenience  of  interpretation,  a 
dot  marks  the  entry  as  it  would  be  made  on  the  graph 
if  the  series  were  taken  as  discrete.  A  slanting  Une  is 
drawn  across  each  class  interval,  beginning  with  the 
summation  total  of  the  preceding  class,  and  ending 


TYPES  AND  MEASURES  OF  DISPERSION      35 

with  the  summation  total  of  the  given  class.  The  given 
frequency  is  thus  represented  as  distributed  evenly- 
through  the  class.  The  resulting  figure  is  known  as 
an  ogive.  To  find  the  quartiles,  the  vertical  scale  rep- 
resenting the  whole  array  is  divided  into  four  equal 
parts,  and  horizontal  lines  are  drawn  from  the  quartile 
division  points  until  the  ogive  is  intersected.    From 


JSO. 

0^     .- 

r 

/%<fj  ///  a  CMnecf/ci/fMi/l,/S9/ 

HO. 

%zoo. 

^^                 _^ 

i 

t 

Y 

Q, 

1  \ 

^ 

j 

50. 

J 

^ 

r   . 

<v' 

jf? 

Hiye: 


^  .JO  1.00  1.50  ZOO  2.50  5.00  5.50  4.00 

Figure  5.     Cumulative  curve,  or  ogive 


the  points  of  intersection  perpendiculars  are  drawn 
to  the  base  line.  The  foot  of  each  perpendicular  marks 
upon  the  horizontal  scale  one  of  the  quartile  values. 
The  deciles  may  be  found  by  dividing  the  horizontal 
scale  into  tenths,  and  proceeding  as  before.  This 
graphic  process,  worked  out  on  large  sheets  of  cross- 
section  paper,  is  usually  the  most  convenient  method 
of  finding  the  quartiles  or  deciles.^ 

*  A   more  complex  form  of   the  ogive  has  recently  been   introduced 
for  testing  the  regularity  of   a  frequency  distribution.      This  ogive   is 


36  INTRODUCTION  TO  ECONOMIC  STATISTICS 

Average  and  Standard  Deviation.  We  shall  now 
consider  the  more  commonly  used  mathematical 
measures  of  dispersion, — the  average  deviation  and  the 
standard  deviation.  The  former  is  coming  to  be  fairly- 
well  known  as  applied  to  economic  data.  The  latter  is, 
however,  generally  favored  by  the  mathematician,  but 
its  chief  statistical  use  at  present  lies  in  connection 
with  the  measurement  of  correlation,  a  subject  which 
will  be  taken  up  later. 

The  principles  involved  in  average  and  standard 
deviation  may  best  be  illustrated  by  taking  a  very 
simple  example.  Suppose  that  four  workers  are  em- 
ployed at  daily  wages  of  $2.00,  $6.00,  $7.00  and  $9.00, 
respectively.  The  average  wage  is  $6.00.  The  first 
wage  differs  from  the  average  by  $4.00,  the  second  is 
at  the  average,  the  third  differs  by  $1.00,  and  the 
fourth  differs  by  $3.00.  The  sum  of  these  differences 
is  $8.00,  or  an  average  of  $2.00  for  each  wage.  The 
average  deviation  (A.  D.)  is  therefore  $2.00,  which  may 
be  taken  as  a  measure  of  the  ''spread"  of  the  wages.^ 
The  standard  deviation  (u)  is  computed  by  squaring 
the  deviations,  averaging  the  squares,  and  finding  the 
square  root  of  this  result.  In  each  case  a  coefficient 
may  be  found  by  dividing  by  the  average  wage.  The 
computations  are  written  out  in  the  following  form : 

drawn  upon  so-called  probability  paper,  and  is  constructed  from  the  sum- 
mated  percentage  frequencies.  The  vertical  scale  of  the  probability 
paper  is  so  graduated  that  a  normal  curve  will  form  a  straight  line 
diagonally  across  the  paper.  The  divergence  of  a  given  distribution 
from  normal  may  be  estimated  by  its  departure  from  a  straight  line. 
The  paper  for  this  graph,  as  well  as  for  other  statistical  work,  may  be 
obtained  from  the  Codex  Rook  Company  of  New  York,  or  from  other 
publishers  of  statistical  material. 

^  The  total  spread,  or  range,  from  the  highest  to  the  lowest  wage  is 
sometimes  given  as  an  inexpert  measure  of  disiJcrsion.  J3ut  it  is  of 
little  value  because  the  wage  limits  are  set  by  single  items  which 
have  only  a  haphazard  relation  to  the  rest  of  the  array.  The  average 
and  standard  deviations,  however,  take  account  of  all  the  items. 


TYPES  AND  MEASURES  OF  DISPERSION      37 


AVERAGE  DEVIATION           j 

STANDARD  DEVIATION 

V 

D 

V 

D 

D» 

$2 
6 
7 
9 

$4 
0 
1 
3 

)  "s 

$2 
6 
7 
9 

$-4 
0 
1 
3 

16 
0 
1 
9 

4  )  24 

4 

4)  24 

0 

4  )   26 

A  =  6 

A.  D. 
Coef. 

=  2 
=^=33% 

A  =  6 

Coef. 

a'  =  6.5 
a  =2.55 

-'f   -43% 

A  practical  application  of  average  deviation  may  be 
cited  from  Dewing,  ''Corporation  Finance,"  Vol.  III. 
The  writer  states  that  the  earnings  of  corporations  pro- 
ducing inexpensive  necessities,  directly  consumed,  are 
most  regular;  while  the  earnings  of  corporations  pro- 
ducing expensive  indirect  goods  are  least  regular.  He 
illustrates  the  two  types  of  corporations  by  the 
Diamond  Match  Company  and  the  American  Locomo- 
tive Company,  computing  the  average  deviations  of 
the  net  earnings  of  the  two  companies.  The  coefficient 
in  the  first  case  is  7.1%,  and  in  the  second  case  50%. 
The  dispersion  might  have  been  measured  in  other 
ways,  as  by  the  standard  deviation,  but  the  quartile 
measure  would  not  be  applicable  to  such  a  small 
number  of  unclassified  deviations. 

The  quartile,  average,  and  standard  deviations  do 
not  give  the  same  results,  as  they  measure  progres- 
sively larger  portions  of  the  frequency  curve  (see  Fig. 
1,  p.  10).  But  in  regular  distributions,  a  comparison 
will  be  the  same  whichever  measure  is  used  to  make 
the  comparison.  In  an  irregular  distribution  which 
has  a  few  extreme  items,  the  standard  deviation  will 
give  an  exceptionally  large  result,  since  the  process  of 
squaring  the  deviations  emphasizes  these  extremes. 

91409 


38  INTEODUCTION  TO  ECONOMIC  STATISTICS 

A  Short-cut  Method.  The  work  of  finding  the  aver- 
age or  standard  deviation  is  often  rather  tedious,  par- 
ticularly when  the  average  is  expressed  as  a  decimal. 
In  finding  the  former,  however,  it  is  not  important  that 
the  average  should  be  expressed  very  precisely,  since 
the  slight  error  involved  in  cutting  short  a  decimal  is 
minimized  in  the  process  of  the  w^ork.  And  in  finding 
the  standard  deviation,  a  short-cut  process  may  be 
used.  In  this  process  a  convenient  average  is  assumed, 
and  a  correction  is  made  later.  The  method  of  making 
the  correction  may  be  illustrated  by  the  simple  wage 
scale  of  four  items  previously  used.  The  average 
wage,  from  which  the  deviations  are  to  be  measured, 
will  be  assumed  to  be  $7.  Needless  to  say,  this  assump- 
tion is  not  here  advantageous,  though  it  would  have 
been  if  the  average  had  been,  let  us  say,  $6.75. 
When  the  deviations  are  measured  from  the  assumed 
average  of  seven,  they  give  an  algebraic  sum  of  — 4, 
which  results  from  the  fact  that  the  ''correction" 
appears  once  in  each  deviation.  Hence  the  algebraic 
sum  of  the  deviations,  divided  by  the  number  of  items, 
will  give  the  ''correction," — the  term  being  taken  to 
mean  the  sum  which  must  be  added  to  the  assumed 
average  to  make  it  the  exact  average.  The  correction 
(K)  when  found  is  added  to  the  assumed  average,  and 
its  square  is  subtracted  from  the  average  squared 
deviation.  By  so  doing,  whatever  error  may  have  been 
involved  in  assuming  an  average  is  eliminated.  Other- 
wise, the  work  is  as  before.  The  computation  is  set 
down  as  indicated  at  the  top  of  the  next  page. 

Deviation  Computed  from  Frequency  Tables.  The 
illustration  that  has  been  considered  thus  far  in  the 


TYPES  AND  MEASURES  OF  DISPERSION      39 


V 
2 
6 

=  7 
9 

D 

—5 

—1 

0 

2 

D2 

25 
1 
0 
4 

4)— 4 

K  =  — 1 

Ax=     7 

4)30 

7.5 

K2  =  l 

A=     6 

a2  =  6.5 
C7  =2.55 

Hoof.  -  ^-^^ 

43% 

discussion  of  average  and  standard  deviation,  has  been 
simplified  by  the  fact  that  the  array  is  not  classified 
into  frequency  groups.  When  computed  from  a  fre- 
quency table,  the  average  and  standard  deviations 
require  a  somewhat  more  complex  process,  though  in 
fact  the  principles  involved  are  precisely  those  already 
explained.  The  one  point  to  be  observed  is  that  the 
vilass  values  must  be  multiplied  by  their  respective 
frequencies  in  order  that  all  items  may  be  taken  into 
account.  The  method  is  shown  in  Table  VII,  where  the 
wage  data  of  1891  are  again  taken  up.^  In  these  com- 
putations, the  class  values  are  taken  at  approximately 
the  mid-points  of  the  class  intervals.  This  is  the  usual 
procedure,  though  a  small  error  is  introduced  by  so 
doing.  An  exact  computation  would  require  that  the 
actual  average  value  of  each  class,  as  determined  by 
dividing  the  total  wages  of  the  class  by  its  frequencies, 
should  be  substituted. 

^A  complex  graph  of  a  frequency  distribution,  known  as  the  Lorenz 
curve,  may  be  described  in  connection  with  the  data  of  Table  VII. 
This  graph   is   based  upon   the  F   and   FV   columns,   as   shown  under 


40  INTRODUCTION  TO  ECONOMIC  STATISTICS 


TABLE  VII 

AVERAGE  DEVIATION  AND  STANDARD  DEVIATION 

WAGE  ROLL  IN  A  CONNECTICUT  MILL,  JULY,  1891 

AVERAGE  DEVIATION 


V 
$  .75 
1.25 
1.75 
2.25 
2.75 
3.25 
3.75 
4.25 


F 

FV 

42 

$31.50 

117 

146.25 

180 

315.00 

10 

22.50 

6 

16.50 

3 

9.75 

1 

3.75 

2 

8.50 

361 

)  553.75 

A  =  1.53 


D 

FD 

5  .78 

$32.76 

.28 

32.76 

.22 

39.60 

.72 

7.20 

1.22 

7.32 

1.72 

5.16 

2.22 

2.22 

2.72 

361) 

5.44 

132.46 

A.D. 

=  .367 
.367 

Coef . = 


1.53 


24% 


Average  Deviation.  The  two  columns   are  reduced  to  percentages  and 
summated,  giving  the  following  results: 
Upper  limit 

of  class  F  (2)                             FV  (2) 

$1.00  11.6%                                5.7% 

1.50  44.0                                    32.1 

2.00  93.9                                   89,0 

2.50  96.7                                   93.1 

3.00  98.3                                  96.1 

3.50  99.2                                    97.8 

4.00  99.4                                   98.5 

4.50  100.0  100.0 

The  two  summated  columns  are  then  plotted  as  coordinates,  the  first  on 
the  horizontal  &eale  and  the  second  on  the  vertical  scale.  If  the  wages 
were  all  alike,  a  direct  diagonal  would  result,  while  disparity  of  wages 
registers  in  the  concavity  of  the  line.  The  use  of  the  five  cent  classes 
would  give  a  more  accurate  representation.  The  curve  has  been  often 
used  for  presenting  a  comparison  of  the  distribution  of  wealth  or  income 
in  difi;ercnt  countries. 


0       20      4a       60       to      too 


TYPES  AND  MEASURES  OF  DISPERSION     41 

STANDAED  DEVIATION   (Assumed  Average  =  $1.75)  ^ 


V 

F 

D 

FD 

FD» 

!  .75 

42 

$-1.00 

$-42.00 

$42.00 

1.25 

117 

-.50 

-58.50 

29.25 

1.75 

180 

0 

0 

0 

2.25 

10 

.50 

5.00 

2.50 

2.75 

6 

1.00 

6.00 

6.00 

3.25 

3 

1.50 

4.50 

6.75 

3.75 

1 

2.00 

2.00 

4.00 

4.25 

2 
361 

2.50 

5.00 

36] 

12.50 

361)-78.00 

1)103.00 

K  =  -.216 

.2853 

Ax  =  1.75 
A  =  1.534 

=    .0467 

=    .2386 

7=    .49 
49 

Formulas.     The  formulas  for  average  and  standard 
deviation  are  as  follows: 

A.  D.  =   — r^   (the  deviations  here  considered  pos- 
itive) 
S.D.  (a)=j/:^_K2 

in  which 

F  =  Frequencies 

D  =  Deviations  (from  assumed  average  if  followed 

by  a  correction) 
N  =  Total  number  of  items  in  array. 
K  =  Correction  for  error  in  assumed  average  = 
SFD 
N 
If  an  average  of  zero  is  assumed,  the  second  formula 
becomes : 


s.D.^f-v!_(-vy 


*A  column  showing  D^  is  often  included,  but  in  most  cases  it  may  bo 
omitted  and  FD^  obtained  by  multiplying  D  x  FD. 


42  INTRODUCTION  TO  ECONOMIC  STATISTICS 

In  some  cases,  particularly  where  a  calculating  ma- 
chine is  used,  this  modification  of  the  formula  will  be 
fouiid  desirable.  It  calls  for  the  computation  of  only 
the  columns  FV  and  FV-.  Its  use  may  be  illustrated 
by  reference  to  Table  IV,  p.  21.  If  the  first  and  third 
columns  are  multiplied  across  and  totaled,  a  result  of 
$890.50  \yi[l  be  obtained.    This  is  SFV^.    Dividing  by 

(2FV\2 
-KJ  1 

gives  $.218,  the  square  root  of  which  is  $.47.  This  is 
the  standard  deviation,  obtained  somewhat  more  accu- 
rately than  before,  since  smaller  class  intervals  are 
taken.  The  modified  formula  will  often  be  found 
useful  in  connection  with  time  series,  where  no  fre- 
quencies are  involved. 

Summary.  In  Table  VIII  a  final  summary  is  made 
of  the  dispersion  of  wages  in  the  Connecticut  mill  here 
studied.  It  is,  of  course,  obvious  that  the  use  of  a 
variety  of  measures  is  for  purposes  of  illustration 
only.  In  practical  work  of  this  sort  only  one  measure 
would  be  used,  probably  either  the  quartile  or  the 
average  deviation.  Skewness  would  doubtless  be  com- 
pared merely  by  an  inspection  of  the  frequency  poly- 
gons. Our  more  exhaustive  study,  however,  gives  a 
very  precise  picture  of  the  dispersion  of  wages.  On 
the  whole,  the  "spread"  lessens  somewhat  after  1870, 
though  the  change  is  not  great  enough  to  render  the 
data  incomparable.  It  will  be  noted  that  the  relative 
skewness  of  the  curves  changes  but  little.  Since,  then, 
the  dispersion  of  wages  does  not  materially  change,  the 
average  wage  may  be  safely  taken  as  an  index  of  the 
periodic  wage  level  in  the  given  mill. 


TYPES  AND  MEASURES  OF  DISPERSION     43 


TABLE  VIII 


DISPEESION  OF  WAGES,  CONNECTICUT  MILL,  JULY,  1870,  1880, 

AND  1891 


MEASURE 


Quartile  Deviation 

Average  Deviation* 

"  "  (exact).. 
Standard  Deviation* 

"  "  (exact).. 
Skewness* 


1870     1880      1891 


$  .21 
.45 
.41 
.62 
.61 
.71 


$  .16 
.28 
.24 
.50 
.46 
.64 


.27 
.37 
.35 
.49 
.47 
.55 


1870     1880      1891 


16% 
31 
30 
44 
44 
114 


14% 
22 
20 
40 
38 
128 


18% 
24 
23 
32 
31 
113 


*  Computed  from  fifty  cent  classes ;  V  =  mid-point  of  class. 

Measurement  of  Skewness.  Tlie  average  deviation, 
as  we  have  seen,  uses  the  first  power  of  the  deviations, 
while  the  standard  deviation  uses  the  second  power. 
If  in  an  analogous  way  the  third  power  is  used,  a 
measure  of  skewness  is  obtained.  A  mathematical 
measure  of  this  sort  is,  however,  seldom  required  in 
economic  statistics.  The  relative  degree  of  skewness 
may  be  roughly  determined  by  comparing  the  outlines 
of  frequency  curves,  or  by  noting  the  position  of  the 
modes  relative  to  the  averages  or  medians.  But  if  an 
accurate  measure  is  desired,  the  following  formula 
should  be  employed : 


Skewness 


f 


N 


The  measure  may  be  reduced  to  a  coefficient  by  divid- 
ing by  the  standard  deviation. 

Library  Work.  The  subjects  of  types  and  disper- 
sion are  treated  in  great  detail  in  several  of  the 
standard  text-books,  such  as  Bowley's  and  Yule's.  The 


44  INTRODUCTION  TO  ECONOMIC  STATISTICS 

most  complete  work  on  the  former  subject  is  that  of 
Zizek,  cited  below.  For  an  exposition  of  graphic  repre- 
sentation, the  student  should  not  fail  to  consult  Brin- 
ton's  work,  and  the  Statistical  Atlas  of  the  United 
States  published  in  connection  with  the  census.  The 
ratio  chart  (semi-logarithmic,  or  *'arith-log,"  paper) 
is  well  treated  in  an  article  by  Irving  Fisher,  as  well 
as  in  an  article  by  J.  A.  Field  reprinted  in  Secrist's 
"Readings."  Whipple's  text-book  gives  an  explana- 
tion of  the  use  of  probabiHty  paper  as  a  means  of 
presenting  and  testing  frequency  curves.  King's  text- 
book, page  156,  explains  the  Lorenz  curve. 

REFERENCES 

Bowley,  Arthur  L.,  Elements  of  Statistics,  Chapters  V-VII. 

Brinton,  W.  C,  Graphic  Methods  for  Presenting  Facts. 

Fisher,  Irving,  "The  Ratio  Chart,"  Quarterly  Publications 
of  the  American  Statistical  Association,  June,  1917,  pp. 
577-601. 

King,  W.  I.,  Elements  of  Statistical  Method,  Chapters  XII- 
XIV. 

Marshall,  Wm.  C,  Graphical  Methods,  Chapters  I-III. 

Secrist,  Horace,  Readings  and  P?-ohlems  in  Statistical  Meth- 
ods, pp.  282-305. 

Whipple,  G.  C,  Vital  Statistics,  Chapter  XII. 

Yule,  G.  U.,  An  Introduction  to  the  Theory  of  Statistics, 
Chapters  VII  and  VIII. 

Zizek,  Franz,  Statistical  Averages. 

EXERCISES 

1.  Using  the  five  cent  frequencies  and  classes,  find  the  aver- 
age wage  for  1870  and  1880. 

2.  Using  25c  class  intervals,  determine  the  mode  for  1870 
and  1880,  following  the  process  illustrated  on  page  24. 

3.  Using  50c  class  intervals,  similarly  determine  the  mode 
for  1870  and  1880. 

4.  Using  the  50c  frequencies,  determine  by  a  mathematical 


TYPES  AND  MEASURES  OF  DISPERSION     45 

formula  the  position  of  the  mode  in  the  1870  and  1880 
data.    Draw  rectangular  histograms  and  smooth  them. 

5.  Explain  why  different  values  for  the  mode  are  obtained 
in  the  two  preceding  exercises.  Which  results  are  the 
more  valid?    Why? 

6.  Find  the  quartile  items  and  their  values  in  the  1870  and 
1880  wage  data  by  interpolating  in  the  5c  classes.  Com- 
pute the  quartile  deviations  and  coefficients. 

7.  Draw  ogives  of  the  wage  data  for  1870  and  1880 — 5e 
frequencies — showing  the  quartile  values, 

8.  Summate  the  percentage  frequencies  from  Table  III,  page 
13,  and  plot  on  probability  paper. 

9.  Find  the  average  deviation  and  coefficient  for  the  1870 
and  1880  wage  data,  50c  classes. 

10.  Find  the  standard  deviation  and  coefficient  for  the  1870 
and  1880  wage  data,  50c  classes,  using  the  method  in- 
volving an  assumed  average  and  correction. 

11.  Using  data  prepared  in  Exercise  9,  draw  Lorenz  curves 
of  wage  distributions  in  1870  and  1880.  Draw  a  similar 
curve  from  the  data  of  Table  IV,  page  21. 

12.  Apply  the  modified  formula  for  standard  deviation  (page 
41)  to  the  five  cent  wage  data  for  1870  and  1880. 

13.  During  a  certain  period  the  rate  of  bank  discount  was 
as  follows : 


Rate  per  cent 

21/2 

3 

No. 

of  days 
174 
408 

31/2 
4 

132 
165 

41/2 
5 

36 
37 

51/2 
6 

20 
26 

7 

2 

(a)  Compute  the  average  rate  of  discount,  taking  the 
number  of  days  as  the  frequencies. 

(b)  In  what  classes  (rate  per  cent)   do  the  quartiles 
fall? 

(c)  What  rate  per  cent  may  be  taken  as  the  mode? 
Why? 

(No  interpolation  is  required  in  the  above  problem.) 
14.    Find  the  coefficients  of  average  deviation,  standard  devia- 
tion, and  skewness  for  the  following  frequency  distribu- 


46  INTRODUCTION  TO  ECONOMIC  STATISTICS 

tion.    Locate  the  quartiles  by  interpolation  and  cheek  the 
results  by  means  of  an  ogive. 

V  F 

$1  1 

2  3 

3  2 

4  2 

5  1 

6  1 

15.  The  following  table  shows  the  increase  in  the  cost  of 
living  for  ten  cities  from  December,  1914,  to  December, 
1920.  Classify  these  percentages  to  the  nearest  multiple, 
of  five,  and  find  the  average  deviation.  (Bureau  of 
Labor  Statistics'  data.) 

Boston    97.4 

Buffalo    101.7 

Chicago    93.3 

Cleveland    104. 

Detroit    118.6 

Los  Angeles    96.7 

New  York   101.4 

Philadelphia    100.7 

San  Francisco 85.1 

Seattle    94.1 

16.  Apply  the  arithmetic  formula  for  determining  the  mode 

to  the  following  three  time  series.  The  months  may  be 
treated  as  if  they  were  class  intervals,  and  the  per- 
centages as  if  they  were  frequencies.  Graph  each  series 
and  construct  a  smoothed  curve : 

Percentage  of  crops  harvested  monthly  in  the  United 
States,  as  reported  by  Department  of  Agriculture. 

Month  Wheat  Corn  Cotton 

May   0.5  —  — 

June    22.0  0.1  — 

July  42.3  0.1  1.4 

Aug 28.4  1.5  11.5 

Sept 6.5  15.8  31.6 

Oct 0.3  28.3  34.4 

Nov —  43.3  16.0 

Dec —  10.9  4.7 

Jan. -April     —  —  0.4 


CHAPTER  III 

INDEXES  OF  WAGES  AND  PRICES 

The  Nature  of  Indexes.  A  large  part  of  statistical 
work  concerns  itself  with  the  making  and  interpreting 
of  indexes.  By  an  index  is  meant  a  number,  whether 
absolute  or  relative,  which  is  used  in  comparisons  to 
measure  a  given  condition.  Used  collectively,  the  term 
implies  a  series  of  such  indexes,  forming  a  multiple 
ratio.  Practically  all  indexes  are  compiled  by  a 
process  of  sampling.  Thus,  though  it  is  impossible  to 
record  any  large  proportion  of  actual  wages  and  prices, 
yet  it  is  possible  to  estimate  changes  in  the  wage  or 
price  level  by  the  use  of  well-selected  samples.  Just 
what  may  be  regarded  as  sufficiently  complete  data  in 
sampling  cannot  be  determined  precisely,  but  must  be 
judged  largely  on  the  basis  of  experience.  Straws 
show  which  way  the  wind  blows,  and  likewise  the  price 
of  a  product  in  a  single  locality  will  often  accurately 
reflect  the  trend  of  a  world  market.  There  is  no 
dependable  uniformity,  however.  Some  prices  respond 
quickly  and  universally  to  changes  in  supply  or  de- 
mand, while  others  move  slowly  and  irregularly.  With 
respect  to  wages,  it  is  commonly  observed  that  the 
market  is  somewhat  slow  in  its  movements.  In  an 
industrial  center  like  New  England,  however,  the 
market  should  be  fairly  responsive.  One  might  never- 
theless hesitate  to  take  the  wages  at  a  single  mill  as 

an  index  of  wages  for  the  whole  country;  but  com- 

47 


48  INTRODUCTION  TO  ECONOMIC  STATISTICS 

parisons  will  show  that  such  an  index  would,  in  fact, 
have  some  dei^ree  of  reliability. 

Indexes  of  Wages.  The  wage  averages  considered 
in  the  preceding  chapter  will  be  taken  provisionally  as 
indexes  of  the  wage  level  in  the  United  States.  Accord- 
ing to  these  indexes,  daily  wages  in  1870  stood  at  $1.38, 
they  fell  by  1880  to  $1.21,  but  climbed  by  1891  to  $1.50. 
The  changes  may  be  presented  more  clearly,  however, 
if  the  figures  are  reduced  to  another  form.  Since  in- 
dexes are  used  as  ratios,  they  may  be  multiplied  or 
divided  through  by  any  factor  to  suit  given  require- 
ments. If  in  this  case  they  are  divided  by  $1.38,  the 
wage  in  1870,  they  are  said  to  be  reduced  to  a  base  of 
1870,  since  the  index  for  that  date  will  become  100.^ 
Expressed  literally,  the  result  is  100%,  but  the  per 
cent  sign  is  usually  dropped  as  being  unnecessary  in 
a  ratio.    The  index  numbers  now  read: 

Year.  Wage. 

1870  100 

1880  88 

1891  109 

Indexes  of  Real  Wages.  In  a  study  of  changes  in 
the  wage  level,  a  further  factor  of  great  importance 
must  be  taken  into  consideration.  This  factor  is  the 
cost  of  living.  Changes  in  the  cost  of  living  affect  the 
prosperity  of  wage  earners  inversely.  Hence  ''real 
wages" — a  term  denoting  the  purchasing  power  of 
wages — will  be  measured  by  money  wages  divided  by 

*  The  base  which  is  theoretically  best  to  use  in  deriving  an  index  from 
absolute  numbers,  is  an  average  of  the  numbers.  Its  advantages  are, 
first,  that  it  is  a  stable  value  from  which  to  measure  the  items,  and 
second,  that  each  index  is  made  to  suggest  its  relative  position  in  the 
series.  Applied  to  the  given  wage  data,  the  base  becomes  the  average 
wage  of  99c,  and  the  indexes  become  101,  89,  and  110 — each  expressing  a 
percentage  of  the  average. 


INDEXES  OF  WAGES  AND  PRICES       49 

prices.  Whether  absolute  or  relative  numbers  are 
taken  to  measure  wages  and  prices,  the  quotients  may 
be  regarded  as  comprising  an  index  of  real  wages,  and 
may  be  reduced  to  any  desired  base.  If  wages  and 
prices  are  expressed  as  indexes  having  the  same  base, 
then  the  resulting  index  of  real  wages  will  also  have 
this  base. 

Various  Wage  Indexes.  The  accuracy  of  our  provi- 
sional wage  index  may  be  tested  and  the  study  ex- 
jtended,  by  the  introduction  of  wage  data  of  a  more 
general  character.  These  data  will  be  taken  from  three 
sources:  (1)  ''The  Movement  of  Wages  in  the  Cotton 
Manufacturing  Industry  of  New  England,"  by  Pro- 
fessor Stanley  E.  Howard;  (2)  The  Aldrich  Report, 
and  (3)  the  publications  of  the  Bureau  of  Labor  Sta- 
tistics, of  the  Department  of  Labor  at  Washington. 
The  first  of  these  sources  gives  a  carefully  prepared 
index  of  weekly  wages  in  the  Massachusetts  cotton 
manufacturing  industry  from  1860  to  1914.  It  is  de- 
rived in  part  from  the  Aldrich  Report,  and  uses  the 
principles  of  tabulation  and  measurement  already 
explained.  The  second  gives  a  general  index  of  wages 
down  to  1891,  based  on  wages  from  many  industries, 
and  covering  various  sections  of  the  country.  The 
third  source  furnishes  an  index  of  hour  rates  in  the 
United  States  from  1840  to  1920.  Hour  rates,  of 
course,  are  not  entirely  satisfactory  as  a  basis  for  an 
index  of  actual  earnings  because  of  the  gradual  reduc- 
tion in  the  length  of  the  working  day.  This  reduction 
has,  however,  been  offset  by  increased  over-time  pay 
at  higher  rates,  by  a  gain  in  leisure  hours,  and  prob- 
ably by  more  regular  employment.  And  as  a  matter 
of  fact,  the  index  of  hour  rates  will  be  found  to  con- 


50  INTRODUCTION  TO  ECONOMIC  STATISTICS 

form  somewhat  closely  to  the  index  of  weekly  wages. 
As  measuring  the  cyclical  changes  in  wages,  both  in- 
dexes give  practically  the  same  results. 

In  Table  IX  the  wage  indexes  from  the  sources  just 
mentioned  are  shown  for  the  years  1870,  1880,  and 
1891,  together  mth  the  indexes  already  derived.  By 
means  of  price  indexes — the  nature  of  which  will  be 
considered  later — nominal  wages  are  reduced  to  real 
wages.    The  indexes  of  both  nominal  and  real  wages 

TABLE  IX 

INDEXES  OF  NOMINAIi  AND  KEAL  WAGES,  JULY,  1870,  1880, 

AND  1891 


SOURCE  OF  DATA 


Connecticut  Mill 

Nominal  wages  (aver- 
age)   

Prices 

Real  wages 

Mass.  Cotton  Mills 

Nominal  wages  (base, 
1860) 

Prices 

Real  wages 

U.  S.-Aldrich  Repprt 

Nominal  wage  (base, 
1860) 

Prices  (base,  1860) .  .  .  . 

Real  wages 

U.  S.  Bureau  of  Lab.  Stat. 

Nominal  wage  (base, 
1913) 

Prices  (base,  1913) . . . . 

Real  wages 


PRIMARY  INDEXES 


1870   1880 


1.37.5 
1.47 
.935 


166 

140 
118 


167.1 
144.4 
116 


67 

147 

46 


1.207 

1.09 

1.11 


154 
105 
147 


143.0 
104.9 
136 


60 

109 

55 


1891 


1.50 

.82 

1.83 


172 

79 

217 


168.6 
94.4 
179 


69 

82 
84 


DERIVED  INDEXES 


1870 


100 

100 

100 
100 

100 

100 

100 

100 


1880 


118 

93 
125 

86 
118 

90 

121 


1891 


109 
196 

104 
184 

101 
154 

103 

185 


are  next  reduced  to  a  base  of  1870,  in  which  form  they 
may  be  readily  compared.  It  will  be  seen  that  the 
indexes  of  real  wages  in  the  Connecticut  mill,  based 
on  very  slender  data  though  they  are,  do  not  ditfer 
markedly  from  the  others. 


INDEXES  OF  WAGES  AND  PRICES        51 

A  more  complete  statement  of  the  results  of  Pro- 
fessor Howard's  study,  and  of  the  Bureau  of  Labor 
Statistics'  index,  is  presented  in  Table  X.    As  in  the 


TABLE   X 
INDEXES  OF  WAGES  AND  PRICES,  1870-1920 


MASSACHUSETTS 

UNITED  STATES 

Year 

Wages 

Wholesale 

Real 

Wages 

Wholesale 

Real 

(Weekly) 

Prices 

Wages 

(Hr.  Rates) 

Prices 

Wageg 

1870 

166 

140 

50 

67 

147 

46 

1871 

177 

130 

58 

68 

136 

50 

1872 

183 

135 

58 

69 

141 

49 

1873 

178 

134 

56 

69 

140 

49 

1874 

163 

128 

54 

67 

133 

50 

1875 

150 

120 

53 

67 

125 

64 

1876 

145 

111 

55 

64 

116 

55 

1877 

142 

108 

55 

61 

113 

54 

1878 

145 

98 

63 

60 

102 

59 

1879 

144 

94 

65 

59 

98 

60 

1880 

154 

105 

62 

60 

109 

55 

1881 

149 

102 

62 

62 

106 

58 

1882 

157 

103 

64 

63 

108 

58 

1883 

158 

98 

69 

64 

102 

63 

1884 

155 

89 

74 

64 

92 

70 

1885 

150 

82 

77 

64 

86 

74 

1886 

153 

80 

80 

64 

84 

76 

1887 

160 

81 

83 

67 

84 

80 

1888 

164 

84 

83 

67 

87 

77 

1889 

169 

81 

89 

68 

84 

81 

1890 

173 

80 

91 

69 

81 

85 

1891 

172 

79 

92 

69 

82 

84 

1892 

172 

75 

97 

69 

76 

91 

1893 

180 

75 

102 

69 

77 

90 

1894 

168 

68 

104 

67 

69 

97 

1895 

165 

66 

105 

68 

70 

97 

1896 

175 

64 

116 

69 

66 

105 

1897 

174 

64 

116 

69 

67 

103 

1898 

171 

66 

110 

69 

69 

100 

1899 

164 

72 

97 

70 

74 

95 

1900 

189 

78 

102 

73 

80 

91 

1901 

190 

77 

104 

74 

79 

94 

1902 

191 

80 

101 

77 

85 

91 

1903 

197 

81 

103 

80 

85 

94 

1904 

196 

80 

103 

80 

86 

92 

1905 

200 

82 

103 

82 

85 

96 

1906 

216 

87 

105 

85 

88 

97 

1907 

240 

92 

HI 

89 

94 

95 

1908 

228 

87 

111 

89 

91 

98 

1909 

211 

90 

99 

90 

97 

93 

1910 

209 

93 

95 

93 

99 

94 

1911 

207 

92 

96 

95 

95 

100 

1912 

223 

95 

99 

97 

101 

98 

1913 

227 

96 

100 

100 

100 

100 

1914 

229 

95 

102 

102 

100 

102 

1915 

(Base 

(Base 

103 

101 

1916 

1860) 

1860) 

111 

124 

1917 

128 

176 

1918 

162 

196 

1919 

184 

(Spring) 

212 

1920 

234 

243 

1 

(Slimmer) 

52   INTRODUCTION  TO  ECONOMIC  STATISTICS 

preceding  table,  an  index  of  real  wages  is  derived  by 
the  use  of  an  index  of  wholesale  prices.  The  price  in- 
dex shown  for  Massachusetts  is  merely  an  adaptation 
of  the  Bureau  of  Labor  Statistics'  data  for  the  United 
States.  The  index  of  real  wages  for  Massachusetts 
has  been  changed  from  a  base  of  1860,  as  first  derived, 
to  a  base  of  1913,  in  order  to  allow  of  comparisons 
with  the  corresponding  index  for  the  United  States. 
Both  indexes  point  to  the  fact  that  real  wages  have 
about  doubled  in  the  interval  from  1870  to  1914,  but 
that  the  greater  part  of  this  increase  came  before  1890. 
Wholesale  and  Retail  Prices.  A  question  may  be 
raised  regarding  the  validity  of  using  an  index  of 
wholesale  prices  as  a  measure  of  changes  in  the  cost 
of  living.  Unfortunately,  no  adequate  index  of  retail 
prices  covering  the  years  here  studied  is  available.  An 
index  of  wholesale  prices  has  therefore  been  substi- 
tuted. It  is  a  well  established  fact,  however,  that 
wholesale  prices  swing  in  the  same  direction  and  at 
nearly  the  same  time  as  retail  prices,  but  that  they 
move  somewhat  more  extremely.  In  the  course  of  the 
usual  moderate  cyclic  changes,  therefore,  the  former 
will  parallel  the  latter  closely  enough  to  serve  as  a 
substitute.  But  when  the  price  swings  are  extremely 
low,  the  substitution  will  doubtless  give  an  exagger- 
ated rise  in  real  wages.  Such  is  evidently  the  case  in 
the  decade  from  1890  to  1900,  when  prices  fell  to  the 
lowest  point  in  the  century.  The  apparent  rise  in  real 
wages  at  that  time  should  therefore  be  discounted, 
though  just  how  much,  cannot  be  accurately  deter- 
mined. The  opposite  result  is  obtained  during  the 
great  upswing  of  prices  from  1914  to  1920.    For  these 


INDEXES  OF  WAGES  AND  PRICES        53 

years,  however,  adequate  data  on  the  cost  of  living 
are  obtainable.  In  Table  XI  estimates  derived  from 
such  data  are  used  in  the  place  of  wholesale  prices, 
and  real  wages  are  then  computed.^ 

TABLE  XI 

INDEXES   OF  WAGES  AND   COST   OF  LIVING 
UNITED  STATES,  1913-1920 

(Estimated  from  Bureau  of  Labor  Statistics'  data) 


Year 

Wages 

Cost  of  Living 

Real  Wages 

1913 

100 

100 

100 

1914 

102 

100 

102 

1915 

103 

100 

103 

1916 

111 

110 

101 

1917 

128 

134 

95 

1918 

162 

154 

105 

1919 

200 

180 

111 

1920 

225 

211 

107 

Indexes  of  the  Cost  of  Living.  The  computation  of 
an  index  of  the  cost  of  living  involves  many  difficulties, 
both  theoretical  and  practical.  To  begin  with,  the  units 
employed  often  vary  in  quality,  and  are  difficult  to 
standardize.  Hence  the  first  step  in  the  computation 
is  the  dramng  up  of  a  selected  list  of  articles  of  staple 
grades.     These  articles  must  be  sufficient  in  number 

^  It  is  not  intended  that  the  wage  data  here  studied  shall  be  taken 
as  a  final  measurement  of  the  course  of  real  wages.  They  were  com- 
piled chiefly  with  the  purpose  in  mind  of  illustrating  certain  methods  of 
work.  But  it  is  the  opinion  of  the  writer,  based  on  a  study  of  such 
material  as  is  available,  that  they  represent  a  passably  good  estimate. 
Other  studies,  however,  have  purported  to  show  a  decline  in  real 
wages  since  the  last  decade  of  the  nineteenth  century.  An  interesting 
study  of  this  sort  appears  in  the  American  Economic  Review,  Septem- 
ber, 1921.  In  this  study  the  cost  of  living  is  assumed  to  be  measured 
by  retail  food  prices.  But  this  is  a  very  questionable  measure,  inas- 
much as  retail  food  prices  have  risen  about  as  rapidly  as  wholesale 
prices  since  the  period  of  agricultural  depression.  This  abnormal  rise, 
of  course,  makes  real  wages  by  comparison  appear  to  fall.  The  study 
also  uses  an  index  of  hour  rates  which  purports  to  show  a  slower 
rise  than  is  shown  by  the  index  published  by  the  Bureau  of  Labor 
Statistics.  It  should  be  noted  that  the  data  here  discussed  do  not  cover 
wages  of  government  employes. 


54   INTRODUCTION  TO  ECONOMIC  STATISTICS 

and  importance  to  represent  fairly  all  the  necessities 
commonly  purchased  by  the  average  working-class 
family.  After  this  has  been  done,  the  prices  of  these 
articles  as  sold  in  representative  stores  must  be  tabu- 
lated. If  the  index  is  to  cover  a  considerable  territory, 
data  must  be  gathered  from  a  number  of  localities. 
The  common  average  of  the  prices  of  each  article  at  a 
given  date  is  found.  Under  certain  conditions,  as 
when  the  extreme  items  are  likely  to  be  in  error,  the 
median  is  preferable  in  place  of  the  common  average. 
Since  the  work  of  collecting  and  tabulating  prices 
requires  elaborate  organization,  it  is  done  on  a  large 
scale  by  only  a  few  agencies.  One  of  these  is  the 
National  Industrial  Conference  Board. ^  Another  is  a 
commission  established  by  the  Legislature  of  the  State 

^  The  National  Industrial  Conference  Board  is  affiliated  with  the  Na- 
tional Association  of  Manufacturers  and  other  similar  organizations, 
and  has  its  headquarters  in  New  York.  It  publishes  among  other  things 
an  excellent  index  of  increases  in  the  cost  of  living  in  the  United 
States.  This  index  is  issued  promptly  each  mouth,  and  is  the  best 
available  source  by  which  current  changes  may  be  measured.  By  per- 
mission, the  indexes  for  the  year  1920  are  here  reprinted.  The  base  is 
July,  1914,  and  the  figures  show  the  per  cent  rise. 


Month, 

Cost  of 

Fuel  and 

1920 

Living 

Food 

Shelter 

Clothing 

Light 

Sundries 

January 

90.2 

97 

43 

170 

49 

77 

February 

93.5 

101 

45 

177 

49 

78 

March 

94.8 

100 

49 

177 

49 

83 

April 

96.6 

100 

50 

188 

51 

83 

May 

101.6 

111 

51 

187 

55 

83 

June 

103.0 

115 

51 

176 

61 

85 

July 

104.5 

119 

58 

166 

66 

85 

August 

103.2 

119 

58 

155 

69 

85 

September 

99.4 

107 

59 

155 

78 

83 

October 

97.3 

103 

59 

148 

83 

90 

November 

93.1 

93 

66 

128 

100 

92 

December 

90.0 

93 

66 

105 

100 

92 

Of  these  indexes  the  only  one  which  shows  a  further  rise  up  to  October, 
1921,  is  that  for  shelter.  This  registers  71%  during  March  to  June, 
inclusive,  and  69%  during  the  four  months  following.  Fuel  and  Light 
and  Sundries  begin  to  decline  after  .lanuary.  Food  shows  an  increase 
for  four  months  following  a  minimum  of  45%  in  June,  but  this  change 
ia  reflected  in  only  a  minor  degree  in  the  aggregate  index.  The  apex 
of  the  post-war  boom  is,  then,  according  to  this  index,  in  July,  1920, 


INDEXES  OF  WAGES  AND  PRICES        55 


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56   INTRODUCTION  TO  ECONOMIC  STATISTICS 

of  Massachusetts.  But  doubtless  the  best  known  and 
most  authoritative  work  is  done  by  the  Bureau  of 
Labor  Statistics.  The  methods  used  in  combining 
average  prices  into  a  single  index  may  be  illustrated 
by  the  Bureau  of  Labor  Statistics'  index  of  the  food 
cost  of  living. 

An  Index  of  Food  Prices.  In  computing  an  index 
of  the  food  cost  of  living,  the  Bureau  of  Labor  Statis- 
tics has  listed  22  important  articles  of  food,  as  shown 
in  Table  XII.  The  use  of  this  list  was  begun  in  Jan- 
uary, 1913,  and  was  continued  to  January,  1921,  when 
a  revised  list  of  43  articles  was  substituted.  Prices  on 
these  articles  were  compiled  monthly  from  a  number 
of  important  cities  selected  from  the  various  terri- 
torial divisions  of  the  United  States.  The  averages 
of  these  prices  for  the  year  1913  and  for  the  month  of 
June,  1920,  are  shown  in  Table  XII,  together  with  the 
annual  consumption  per  workingman's  family  as  ascer- 
tained by  a  special  investigation,  which  is  here  as- 
sumed to  apply  directly  to  1913.  From  these  data  the 
increase  in  the  food  cost  of  living  from  the  pre-war 
level  to  the  climax  of  prices  in  1920  may  be  computed. 

The  Aggregate  Method.  The  two  standard  methods 
of  making  the  computation  are  shown  in  Table  XII. 
The  first  and  simplest  of  these  is  known  as  the  aggre- 
gate method,  and  is  illustrated  in  the  first  five  columns 
of  the  table.  The  process  consists  merely  in  finding 
the  total  cost  of  a  year's  supply  of  the  given  articles 
at  the  two  contrasted  dates  (columns  3  and  5.)  The 
total  cost  in  June,  1920  ($479.44),  is  then  divided  by 
the  total  cost  in  1913  ($215.89).  The  result  shows  that 
the  former  cost  was  222%  as  compared  with  100%  in 
the  base  year,  1913 — an  increase  of  122%.    While  the 


INDEXES  OF  WAGES  AND  PRICES        57 

totals  do  not  actually  represent  more  than  about  two- 
thirds  of  the  average  annual  food  cost  per  family,  yet 
they  doubtless  are  representative  enough  to  give  a 
close  approximation  to  the  correct  percentage  increase. 

The  Proportional  Expenditure  Method.  A  second 
and  more  complex  process,  known  as  the  proportional 
expenditure  method,  is  illustrated  in  columns  6,  7,  and 
8.  By  this  method  the  relative  price  in  June,  1920,  as 
compared  with  1913  is  found  (column  6).  By  so  doing, 
all  prices,  whether  large  or  small,  are  obviously  put 
on  the  same  basis,  since  each  is  based  on  100%  in  1913. 
These  relative  prices  are  next  averaged  by  a  process  of 
weighting  similar  to  that  explained  in  Chapter  II  in 
connection  with  Table  IV.  The  weights,  as  shown  in 
column  7,  measure  the  relative  importance  in  a  work- 
ingman's  budget  of  an  annual  supply  of  each  article. 
To  illustrate,  the  first  weight  is  obtained  by  dividing 
$8,128  by  $215.89.  This  gives  approximately  3.8%,  but 
both  the  decimal  point  and  the  per  cent  sign  are 
dropped  as  being  unnecessary  in  a  ratio.  In  the  same 
way  the  second  weight,  33,  is  obtained  by  dividing 
$7,136  by  $215.89,  and  so  on  for  the  remainder  of  the 
weights.  The  accuracy  of  the  work  may  be  checked 
by  means  of  the  total,  which  should  come  to  approxi- 
mately 1000.  In  the  computation  of  such  weights,  it 
is  usually  unnecessary  to  strive  for  extreme  accuracy ; 
as,  for  example,  by  carrying  the  figures  to  several 
decimal  places.  A  considerable  margin  of  error  may 
be  present  in  the  weights  without  materially  affecting 
the  weighted  average. 

A  Comparison  of  the  Two  Methods.  It  will  be  ob- 
served that  the  result  by  the  proportional  expenditure 
method  is  exactly  the  same  in  this  instance  as  that 


58    INTRODUCTION  TO  ECONOMIC  STATISTICS 

obtained  by  the  aggregate  method.  But  this  would  not 
always  be  the  case.  Where  there  is  a  difference,  pref- 
erence is  likely  to  be  given  to  the  result  obtained  by  the 
former  method.  The  jid vantage  claimed  for  this 
method  arises  from  the  fact  that  it  is  not  practicable 
to  make  frequent  revisions  of  the  consumption  esti- 
mates. When  an  estimate  has  become  somewhat  out 
of  date,  its  inaccuracies  may  have  considerable  effect 
upon  the  results  obtained  by  the  aggregate  method. 
But  it  is  assumed  that  the  proportional  expenditure 
weights,  based  as  they  are  upon  both  the  prices  and 
the  consumption  of  a  specific  period,  will  remain  rela- 
tively accurate  for  a  longer  period  than  will  the  con- 
sumption estimates  taken  alone.  Hence  their  use  is 
thought  to  give  somewhat  more  dependable  results. 
In  practice  it  will  be  found  that  the  proportional  ex- 
penditure method  is  not  as  tedious  as  it  seems  in  the 
illustration.  Tlie  weights,  when  once-  obtained,  may 
be  used  unchanged  for  a  long  period;  and  the  finding 
of  the  relative  prices  does  not  usually  mean  additional 
work,  since  they  are  in  any  case  desirable  for  purposes 
of  comparison. 

The  formuia  for  the  price  index  by  the  aggregate 
method  is : 

p  _2pnqx 

X  n  —  XT' 

^Poqx 
in  which 

Pn  =  price  index  for  the  relative  period 

Pn  =  prices  for  the  relative  period 

q,  =  quantities    consumed    during   a   given   period, 

preferably  the  base  period 
Po  =  prices  for  the  base  period 
The  process  may  be  described  briefly  as  a  comparison 


INDEXES  OF  WAGES  AND  PRICES        59 

of  two  price  averages,  both  obtained  by  woighiiiig  for 
quantities. 

The  formula  for  a  price  index  by  (he  proportional 
expenditure  method  is  :^ 

SP^XpxQx 
JTn  —         po 

SpxqT" 

This  process  may  be  described  as  an  average  of  rela- 
tive prices  weighted  for  the  vahie  eoiisuuied  during 
some  specified  period,  often  prior  to  the  beginning 
of  the  interval  over  which  the  price  comparison  is  be- 
ing made.  In  connection  with  both  formulas,  it  must 
be  understood  that  the  factors  are  taken  distributively  ; 
that  is,  only  prices  and  weights  belonging  to  the  same 
article  are  multiplied.'^* 

An  inspection  of  the  fornmlas  will  show  why  the 
two  results  obtained  in  Table  XII  are  alike.  As  is 
often  the  case,  the  year  indicated  by  the  subscript  x, 
to  which  the  quantities  and  weights  are  assumed  to 
belong,  is  identical  with  the  base  year  indicated  by  the 

'  The  formula  as  thus  stated  docs  not  indicato  tlio  reduction  of  tlio 
proj)ortional  oxpendituro  weijrlits  to  ])c'rfi>ntagos.  Tlio  valuo  is,  of 
course,  unaircctod  by  audi  loduction;  and  for  j)urj)oflcs  o*'  lator  com- 
parison tlio  form   as  jj^iven  is  proforablo. 

'Use  is  somotimofl  made  of  tho  Imrmoiiic  mean  In  Hiidin^j  tlie  average 
of  the  relatives.  This  averaj:je  is  found  1/y  lakinp  the  reciprocals  oJ 
the  relatives,  comjjutinfj  their  averajje,  and  then  takiii}^  the  reciprocal 
of  this  averaj^'e.  it  may  easily  bo  seen  that  the  effect  of  takiiifj^  the 
reciprocals  of  the  relatives  is  to  reverse  the  base;  that  is,  tlu!  later  year 
becomes  the  bast<  instead  of  the  earlier  year.  If  vveij^hts  nrc^  usi-d  it  is 
therefore  preferable  that  they  be  derived  from  the  data  of  the  lat.i^r 
year  in  (indinjf  the  harmonic  mean.  Takin;j  the  reciprocal  of  the 
averajife  again  reverses  the  bases.  Rut  the  index  obtained  by  the  usual 
direct  method  will  not  be  quite  the  same  as  that  obtained  by  the 
harmonic  mean.  This  will  ordinarily  be  the  case  whether  the  weif^hta 
are  shifted  to  the  later  base,  or  not;  or  indee<l  whether  any  weights 
are  employed  or  not.  The  same  distinction  between  the  arithmetic  and 
the  harmonic  mean  is  well  brought  out  by  taking  a  flimple  average 
of  a  given  set  of  prices,  and  comparing  it  with  tho  harmonica  mean  of 
the  same  prices.  The  latter  is  the  recii)rocal  of  the  average  of  the 
quantities  of  each  commodity  purchasable  for  ono  dollar. 


60  INTRODUCTION  TO  ECONOMIC  STATISTICS 


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INDEXES  OF  WAGES  AND  PRICES        61 

subscript  o.  The  numerator  of  the  second  formula 
therefore  reduces  to  the  same  form  as  that  of  the  first 
formula,  and  the  denominators  also  have  the  same 
value.  But  the  assumption  that  the  quantities  as  given 
apply  directly  to  the  year  1913  was  adopted  merely  to 
simplify  the  table  for  purposes  of  illustration.  As  a 
matter  of  fact,  these  quantities  were  obtained  by  an  in- 
vestigation made  in  1918,  and  were  not  put  into  use 
until  the  beginning  of  1921.  The  consumption  esti- 
mates actually  used  by  the  Bureau  of  Labor  Statistics 
from  1913  to  1920,  inclusive,  are  shown  in  Table  12-A, 
together  with  the  annual  retail  food  prices  for  the 
same  period,  and  for  June,  1921.  It  will  be  seen  that 
these  estimates  go  back  to  the  year  1901. 

Limitations.  Some  of  the  limitations  of  the  fore- 
going methods  of  computing  changes  in  the  cost  of  liv- 
ing may  be  easily  seen.  It  is  evident  that  the  assum- 
ing of  an  unvarying  consumption  may  sometimes  result 
in  material  inaccuracies.  If  there  is  a  very  uneven 
rise  in  prices,  buyers  will  begin  to  substitute  the 
cheaper  article  for  the  more  expensive,  wherever  this 
can  be  properly  done.  The  effect  on  the  one  hand 
will  be  to  moderate  the  unevenness  of  price  changes, 
but  on  the  other  hand  it  will  make  material  altera- 
tions in  the  budget.  But  the  frequent  revision  of  con- 
sumption data,  and  the  complexity  of  methods  of  com- 
putation, would  involve  more  labor  than  at  present 
seems  practicable  to  allow.  Besides,  the  revision  of 
consumption  data  raises  another  problem.  Changing 
quantities  may  in  part  reflect  rising  standards  of  com- 
fort whereas  the  term,  ''cost  of  living,"  implies  pro- 
vision only  for  those  necessities  which  are  required  to 
maintain  the  social  efficiency  of  the  family.    Strictly 


62  INTRODUCTION  TO  ECONOMIC  STATISTICS 

speaking,  the  measurement  of  changes  in  such  costs 
would  call  for  the  dietician 's  units  of  protein,  carbohy- 
drates, fats,  and  calories,  and  would  require  very  de- 
tailed analyses.  Hence,  as  far  as  the  cost  of  living 
is  concerned,  we  are  driven  back  to  the  comparatively 
simple  methods  actually  in  use.  The  results  obtained 
by  such  methods  must,  of  course,  always  be  taken  as 
approximations  only. 

Combining  the  Partial  Indexes.  In  measuring 
changes  in  the  entire  cost  of  living,  the  Bureau  of 
Labor  Statistics  makes  an  index  for  several  groups  of 
commodities,  in  addition  to  food.  The  process  of  com- 
bining these  partial  indexes  into  a  single  measure  is  an 
application  of  the  proportional  expenditure  method. 
The  budgetary  studies  of  1918  furnish  the  weights 
now  in  use.  The  partial  indexes  in  June,  1920,  the 
weights,  and  the  process  of  finding  a  single  index,  may 
be  seen  in  the  following  table : 

Index  Weight 

Item  of  June,  1920      (percent 

Expenditure  (base,  1913)   of  budget)    Extension 

Food    219.0  38.2  8365.80 

Clothing   287.5  16.6  4772.50 

Housing    134.9  13.4  1807.66 

Fuel  and  light 171.9  5.3  911.07 

Furniture  and 

furnishings  292.7  5.1  1492.77 

Miscellaneous    201.4  21.3  4289.82 

99.9     )     21639.62 

216.5 

Indexes  of  Wholesale  Prices.     Though  neither  the 

aggregate  nor  the  proportional  expenditure  method  is 

theoretically  perfect,  yet  one  or  the  other  is  used  in 

practically  all  the  price  indexes  in  common  use.     In 


INDEXES  OF  WAGES  AND  PRICES        63 


adapting  the  latter  method  to  wholesale  prices,  the 
weights  are  usually  made  to  reflect  the  proportional 
value  produced,  rather  than  family  consumption,  but 
the  principle  is  essentially  the  same.  Applying  this 
method,  the  Bureau  of  Labor  Statistics  compiles  an 
elaborate  monthly  index  of  wholesale  prices.  The 
number  of  items  tabulated,  the  base  used,  and  other 
details  of  the  work  have  been  varied  from  time  to 
time,  but  the  partial  results  have  been  combined  into  a 
single  index  having  the  same  base.  For  several  years 
prior  to  the  war,  the  base  used  was  an  average  of  the 
decade  1890-99,  which  was  a  period  of  unusually  low 
prices.  But  in  more  recent  years,  1913  has  been  em- 
ployed because  it  may  be  considered  representative 
of  normal  conditions  immediately  preceding  the  war. 
As  compiled  for  January,  1920,  the  index  is  derived 
from  a  weighted  average  of  327  price  quotations. 
The  items  are  classified  under  nine  headings,  each 
having  its  own  partial  index.  ^    This  is  the  most  com- 

^  Annual  group  index  numbers  for  the  years  1890-1920  will  be  found 
in  the  Monthly  Labor  Beview,  Feb.,  1921,  page  45.  The  figures  for 
recent  dates  are  as  follows: 

GEOUP  INDEX  NUMBERS— UNITED  STATES— BUREAU  OF 

LABOR  STATISTICS 

For  the  Years  1913-1920,  and  Sept.,  1920  and  1921 


•a 

•o 

■a 

a 
.So 

1| 

P  bo 

3Sa 

.2  C 

aj2 

1913 

100 

100 

100 

100 

100 

100 

100 

100 

100 

1914 

103 

103 

98 

96 

87 

97 

101 

99 

99 

1915 

105 

104 

100 

93 

97 

94 

114 

99 

99 

1916 

122 

126 

i-ZH 

119 

148 

101 

159 

115 

120 

1917 

189 

176 

181 

175 

208 

124 

198 

144 

155 

1918 

220 

189 

239 

163 

181 

151 

221 

196 

193 

1919 

234 

210 

261 

173 

161 

192 

179 

236 

217 

1920 

218 

239 

302 

238 

186 

308 

210 

366 

236 

Sept. 

1920 

210 

223 

278 

284 

192 

318 

222 

371 

239 

Sept. 

1921 

122 

146 

187 

178 

120 

193 

162 

223 

146 

64  INTRODUCTION  TO  ECONOMIC  STATISTICS 

prehensive  study  of  wholesale  prices  published,  but 
for  business  uses  it  has  the  disadvantage  of  appearing 
two  or  three  months  late. 

One  of  the  most  widely  used  commercial  indexes  of 
wholesale  prices  is  Bradstreet's.  This  index  is  based 
on  about  one  hundred  important  articles,  for  which 
prices  are  easily  available  in  the  principal  business 
centers.  It  is  published  promptly  each  month,  and 
is  therefore  valuable  to  the  business  man  who  desires 
to  keep  in  touch  with  the  immediate  trend  of  the  mar- 
ket. The  aggregate  method  of  computation  is  em- 
ployed, and  the  result  is  given  merely  as  a  sum  of 
money,  which  may  be  reduced  to  any  desired  base. 
The  lack  of  a  definite  system  of  weights  is  a  defect 
of  this  index,  but  it  is  partially  compensated  for  by  a 
careful  selection  of  items,  and  a  repetition  of  important 
articles  by  a  quotation  for  more  than  one  grade. 

Dun's  monthly  index  of  wholesale  prices  is  perhaps 
not  as  widely  known  as  Bradstreet's,  but  it  appears  to 
be  more  scientifically  compiled.  It  is  based  upon  ap- 
proximately 300  price  quotations,  which  are  grouped 
under  seven  heads,  and  finally  combined  into  a  single 
index,  the  result  being  stated  in  dollars.  Prices  are 
weighted  in  accordance  with  estimated  per  capita  con- 
sumption. Just  how  the  weights  are  applied  is  not 
apparent.  Commercial  houses,  as  a  rule,  do  not  pub- 
lish the  details  of  their  statistical  work. 

The  Federal  Reserve  Board  has  begun  the  publica- 
tion of  a  monthly  index  of  wholesale  prices,  based  on 
about  90  commodities.  It  is  intended  for  use  particu- 
larly in  international  comparisons,  but  its  value  is 
not  limited  to  this  field.'    Still  another  index  intended 

^  The  Federal  Eeserve  Board  classifies  its  data  so  as  to  present  indexes 


INDEXES  OF  WAGES  AND  PRICES        65 

to  measure  wholesale  price  movements  is  published  by 
the  Babson  Statistical  Organization.  This  index  is 
compiled  monthly  from  quotations  on  ten  basic  com- 
modities, and  in  spite  of  its  narrow  base,  serves  a  use- 
ful purpose.  A  number  of  other  less  complete  indexes 
might  be  mentioned,  some  of  which  appear  weekly. 
One  of  these  is  the  Annalist's  weekly  index  of  whole- 
sale food,  a  weighted  average  of  25  prices.  It  is  obvious 
that  there  may  be  many  price  indexes  giving  somewhat 
different  results,  yet  each  one  valid  in  its  own  sphere. 
The  purpose  a  given  index  is  to  serve  will  always  con- 
trol the  selection  of  items  and  the  weighting. 

As  a  means  of  comparing  some  of  the  indexes  just 
mentioned,  Table  XIII  is  given,  showing  several  whole- 
sale price  indexes  for  the  year  1913,  and  by  months 
for  1920.  The  data  for  1920  are  of  interest  because 
they  contain  the  peak  of  prices  for  the  war  and  post- 
war cycle. ^ 

TABLE  XIII 
INDEXES  OF  WHOLESALE  PEICES 


Bureau 

Brad- 
street  's 

Federal 

Period 

of  Labor 

Dun's 

Reserve 

Babson 's 

Statistics 

Board 

1913 

100 

$9.2115 

$120.8865 

100 

$1.26 

1920 

Jan. 

248 

20.3638 

247.394 

242 

3.30 

Feb. 

249 

20.8690 

253.748 

242 

3.44 

Mar. 

253 

20.7950 

253.016 

248 

3.59 

Apr. 

265 

20.7124 

257.901 

263 

3.61 

May 

272 

20.7341 

263.332 

264 

3.66 

June 

269 

19.8752 

262.149 

258 

3.71 

July 

262 

19.3528 

260.414 

250 

3.60 

Aug. 

250 

18.8273 

252.288 

234 

3.43 

Sept. 

242 

17.9746 

248.257 

226 

3.39 

Oct. 

225 

16.9094 

237.341 

208 

3.25 

Nov. 

207 

15.6750 

227.188 

190 

2.98 

Dec. 

189 

13.6263 

211.628 

171 

2.75 

for  goods  imported  and  exported,  and  producers'  and  consumers'  goods, 
as  well  as  for  certain  commodity  groups.  These  indexes  are  published 
monthly  in  the  Federal  Beserve  Bulletin. 

^  In  comparing  these  data,  it  should  be  noted  that  the  commercial 
indexes  (Bradstreet 's.  Dun's,  and  Babson 's)  are  based  on  price  quota- 
tions for  the  first  of  each  month. 


66  INTRODUCTION  TO  ECONOMIC  STATISTICS 

Other  Indexes.  In  addition  to  indexes  of  commodity 
prices,  many  other  indexes,  measuring  various  phases 
of  business  activity,  are  published.  Speculative  and 
investment  activities  are  measured  by  indexes  of  stock 
and  bond  prices.  A  good  index  of  stock  prices  is  dif- 
ficult to  make  because  of  the  continually  changing 
status  of  the  corporations  issuing  the  stocks.  The 
indexes  now  in  common  use  are  based  on  quotations  of 
standard  shares  listed  on  the  New  York  Stock  Ex- 
change. Railroad  and  industrial  shares  are  usually 
compiled  separately,  and  the  two  sets  combined  into  a 
composite  index.  Financial  and  productive  activities 
in  the  general  markets  are  measured  by  a  variety  of 
data,  such  as  the  number  of  shares  traded  on  the  New 
York  Stock  Exchange,  bank  clearings  in  New  York 
and  in  the  country  as  a  whole,  bank  deposits,  money 
in  circulation,  gold  movements,  the  interest  rate,  the 
production  of  pig  iron,  the  number  of  building  per- 
mits issued  in  leading  cities,  the  number  and  extent  of 
business  failures,  and  the  balance  of  foreign  trade. 
Labor  conditions  are  measured  by  wage  and  employ- 
ment data,  examples  of  which  are  the  reports  of  the 
New  York  State  Industrial  Commission.  Retail  trade 
is  indicated  by  reports  of  department  store  sales. 
Most  of  these  various  indexes  as  they  appear  in  the 
financial  papers,  consist  of  mere  statements  of  periodi- 
cal figures.  Certain  phases  of  their  statistical  elabora- 
tion will  be  considered  later  under  the   subject   of 

trends. 

REFERENCES 

Bamett,  George  E.,  "Index  Numbers  of  the  Total  Cost  of 
Living,"  Quarterly  Journal  of  Economics,  February,  1921, 
pp.  240-263. 


INDEXES  OF  WAGES  AND  PRICES       67 

Bowley,  Arthur  L.,  "The  Measurement  of  Changes  in  the 
Cost  of  Living"  (and  discussion),  Journal  of  the  Royal 
Statistical  Societij,  May,  1919,  pp.  343-372. 

Fisher,  Irving,  Stahilizing  the  Dollar,  Chapters  I-III. 

Howard,  Stanley  E.,  The  Movement  of  Wages  in  the  Cotton 
Manufacturing  Industry  of  New  England. 

Meeker,  Royal,  ''Some  Features  of  the  Statistical  Work  of 
the  Bureau  of  Labor  Statistics,"  Quarterly  Publications  of 
the  American  Statistical  Association,  March,  1915,  pp.  431- 
441. 

Mitchell,  Wesley  C,  Index  Numbers  of  Wholesale  Prices  in 
the  United  States  and  Foreign  Countries,  Bulletin  No.  173 
(Whole  Number),  Bureau  of  Labor  Statistics,  July,  1915. 

Secrist,  Horace,  An  Introduction  to  Statistical  Methods, 
Chapters  IX  and  X. 

Secrist,  Horace,  Readings  and  Problems  in  Statistical 
Methods,  Chapter  VIII. 

Stewart,  Walter  W.,  "Prices  During  the  War,"  Quarterly 
Publications  of  the  American  Statistical  Association,  Sep- 
tember, 1920,  pp.  305-313. 


EXERCISES 

1.  Reduce  the  three  indexes  shown  on  page  53  to  a  base 
consisting  of  the  average  of  each  series,  respectively. 

2.  Reduce  Bradstreet 's.  Dun's  and  Babson's  indexes  of 
wholesale  prices  for  1920  (page  65)  to  a  1913  base. 
Compare  the  divergence  of  these  indexes,  and  the  others 
given  in  the  same  table,  for  the  month  of  May  by  the 
use  of  the  coefficient  of  average  deviation. 

3.  (a)  Plot  on  the  same   sheet  of  semi-logarithmic  paper  the 
two  indexes  of  real  wages  given  in  Table  X,  page  51. 
(b)    Plot   together   on   the   same   chart   the   indexes   of 
wages  and  cost  of  living  shown  in  Table  XI,  page  53. 

4.  Assuming  that  the  ratio  of  American  to  European  com- 
modity prices — both  price  levels  being  stated  in  indexes 
having  1913  as  a  base — measures  the  relative  deprecia- 
tion of  European  currencies,  find  the  theoretical  value 
of  these  currencies  for  September,  1921,  using  the  data 
given    below    {Federal  Reserve   Bulletin,   Nov.,    1921). 


68  INTRODUCTION  TO  ECONOMIC  STATISTICS 

Express  the  market  prices  of  exchange  as  percentages 
of  the  theoretical  values. 

Exchange  in 
Index  of  New  York. 

Wholesale  Prices     (Cables,  per 
Country  September,  1921     cent  of  par) 

United  States 143  — 

United  Kingdom  ...  191  76.5 

France     344  37.7 

Italy     580  21.8 

Germany     1777  4.0 

Sweden    182  81.3 

Norway    287  48.0 

Denmark 224  65.9 


5.  The  following  table  gives  a  monthly  index  of  wholesale 
prices  obtaining  in  the  United  Kingdom  ("Statist"  in- 
dex) and  the  average  monthly  price  in  New  York  of 
sterling  exchange  (cables),  for  the  year  1920.  Using  the 
Federal  Reserve  Board  index  (page  65)  as  a  measure 
of  the  price  level  in  the  United  States,  find  the  theoretical 
value  of  the  pound  sterling  (par,  $4.8665)  each  month 
in  terms  of  American  money.  Compare  the  results  thus 
obtained  with  the  cost  of  sterling  exchange  by  graphing 
both  series.  (The  graph  should  be  drawn  so  as  to  show 
the  zero  line  at  the  base.     Why?) 

Statist  Sterling 

index  cables 

1920                                (Base,  1913)  (New  York) 

January     288  $3.68 

February    306  3.39 

March    307  3.72 

April    313  3.93 

May    305  3.85 

June   300  3.95 

July    299  3.86 

August    298  3.63 

September    292  3.52 

October    282  3.47 

November    263  3.43 

December    243  3.63 


INDEXES  OF  WAGES  AND  PRICES        69 

6.  From  the  data  of  Table  XII-A,  page  60,  find  index  num- 
bers of  the  food  cost  of  living,  1913-1920,  inclusive,  and 
June,  1921 ;  base,  1913.  Compare  the  results  with  the  fol- 
lowing index  published  by  the  Bureau  of  Labor  Statistics : 

Index  of 
Year  food  costs 

1913 100 

1914 102 

1915 101 

1916 114 

1917 146 

1918 168 

1919 186 

1920 203 

June,  1921 144* 

*  Based  on  43  articles. 

7.  From  the  average  prices  given  below,  find  the  relative 
prices  in  1918  as  compared  with  1913.  Find  also  the 
weighted  average  of  these  "relatives,"  applying  the 
weights  given. 

1913  1918  Weights 

Wlieat,  bushel   $1.04  $2.31              4 

Corn,  bushel 71  1.84            10 

Cotton,  pound 13  .32              5 

Iron,  ton 14.90  36.52               3 

Copper,  pound 16  .25              1 

8.  On  the  basis  of  the  following  prices  of  five  articles  of 
food,  and  the  relative  importance  of  each  in  the  family 
budget,  find  the  increase  in  the  food  cost  of  living  be- 
tween the  dates  given : 

Price  in  Price  in 

Article  1913  1920  Importance 

Steak,  pound   $0.22  $0.40  70 

Milk,  quart 09  .17  140 

Bread,  pound 06  .12  140 

Butter,  pound 38  .70  115 

Sugar,  pound 06  .20  40 

9.  Compute  a  set  of  proportional  expenditure  weights  on 
the  basis  of  1901  consumption  and  1913  prices.  Apply 
these  weights  to  the  relatives  given  in  column  6,  Table 
XII,  page  55. 


70  INTRODUCTION  TO  ECONOMIC  STATISTICS 

10.  From  the  following  table,  compute  an  index  of  agricul- 
tural real  wages  having  1913  as  its  base.  Compare  the 
results  with  the  corresponding  index  of  hour  rates  in 
the  United  States  by  graphing  both  on  the  same  chart 
for  the  years  in  which  farm  wages  are  given.  (The  base 
line  of  the  graph  should  show  each  year  from  1875  to 
1920.    Why?) 


WAGES  OF  CERTAIN  CLASSES  OF  MALE  FARM  LABOR  BY  THE 
MONTH  WITHOUT  BOARD 

("Monthly  Labor  Review,"  July,  1920,  and  March,  1921.) 

Year  Wage 

1875   19.87 

1879    16.42 

1882    18.94 

1885    17.97 

1888    18.24 

1890    18.33 

1892    18.60 

1893    19.10 

1894    17.74 

1895    17.69 

1898    19.38 

1899    20.23 

1902    22.14 

1910  27.50 

1911  28.77 

1912  29.58 

1913  30.31 

1914  29.88 

1915  30.15 

1916  32.83 

1917  40.43 

1918 48.80 

1919  56.29 

1920  64.95 

11.  The  Bureau  of  Labor  Statistics  found  the  increases  in 
the  cost  of  living  for  various  classes  of  commodities  in 
the  United  States,  from  1913  to  the  year  and  month 
indicated  to  have  been  as  follows: 


INDEXES  OF  WAGES  AND  PRICES  71 

Per  Cent  of  Increase 

Item  of                   Dec.  Dec.     Dec.     Dec.     Dee.  Dec.  Dec. 

Expenditure                1914  1915    1916     1917    1918  1919  1920 

Food    5.0       5.0     26.0     57.0       87.0  97.0  78.0 

Clothing    1.0       4.7     20.0     49.1     105.3  168.7  158.5 

Housing     0.0       1.5       2.3         .1         9.2  25.3  51.1 

Fuel   and   light 1.0       1.0       8.4     24.1       47.9  56.8  94.9 

Furniture  and 

furnishings 4,0     10.6     27.8     50.6     113.6  163.5  185.4 

Miacellaneous     3.0       7.4     13.3     40.5       65.8  90.2  108.2 


Using  the  weights  given  on  page  62,  find  the  total  in- 
crease in  the  cost  of  living,  and  the  index  of  the  same, 
for  the  periods  named  in  the  foregoing  table.  Compare 
the  results  with  those  given  in  the  Monthly  Labor  Re- 
view, November,  1921,  page  83. 
12.  Using  price  quotations  obtained  locally,  and  from  cata- 
logs from  which  local  purchases  are  commonly  made, 
find  the  increase  in  the  cost  of  living  between  two  given 
dates,  preferably  1913  or  1914  and  the  present  time.  In 
finding  the  food  index,  make  use  of  the  proportional  ex- 
penditure weights  given  in  Table  XII,  page  55.  For 
the  other  groups  of  items,  make  use  of  the  following  pro- 
portional expenditure  weights  quoted  from  Massachusetts 
House  Report,  No.  1500.  Items  for  which  quotations  are 
not  available  may  be  dropped,  or  substitutions  may  be 
made. 


WEIGHTINGS  IN  THE  CLOTHING  INDEX 
MEN'S 

Overcoat! 

Suit         \ 39 

Trousers  J 

Shoes   15 

Hats    4 

Gloves    6 

Socks   4 

Shirts 6 

Collars 2 

Underwear    6 

Night   Garments    2 

Total 84 


72  INTRODUCTION  TO  ECONOMIC  STATISTICS 

WOMEN 'S 

Suit  ] 

Topcoat         y  •  • 27 

Street  DressJ 

Underwear 5 

"Waists 
Kimono 

House  Dress  |^ 18 

Aprons 
Night  Gown 
Underskirt 

Shoes : 12 

Gloves    3 

Hosiery   2 

Corsets 4 

Hats    9 

Total 80 

SHELTER  INDEX 

Obtain  rentals  of  several  representative  homes  for  the 

two  dates  required,  and  find  the  average  per  cent  in- 
crease. 

WEIGHTINGS  IN  THE  FUEL  INDEX 

Coal   10 

Kerosene 1 

Gas 2 

Electricity 2 

Total 15 

WEIGHTINGS  IN  THE  SUNDRY  INDEX 

Ice    15 

Carfare 15 

Entertainment    25 

Medicine    25 

Insurance    50 

Church    30 

Tobacco,  etc 20 

Reading 10 

House  furnishings  45 

Organizations 25 

Total 260 


INDEXES  OF  WAGES  AND  PRICES        73 

Combine  the  various  partial  indexes  by  means  of  thp 
following  proportional  expenditure  weightsf 

Food   43  J 

Shelter    * '  '  jy"™ 

Clothing ' '  * '  232 

Fuel  and  light *.*.'.'.*.*.'."*     56 

Sundries .'!.*!!!!'  20*4 


CHAPTER  IV 
QUANTITY  INDEXES  AND  THEIR  USES 

Value  and  Quantity  Indexes.  Students  of  general 
market  conditions  are  interested  in  changes  in  the 
volume  of  production,  since  such  changes  have  an  inti- 
mate relation  to  prices,  wages,  and  business  activity. 
The  attention  of  statisticians  has  therefore  been  re- 
cently turned  to  the  development  of  indexes  of  pro- 
duction. Such  indexes  may  be  of  two  sorts;  (1)  an 
index  of  value  production,  measuring  the  number  of 
dollars'  worth  of  goods  created  in  a  given  period  of 
time;  and  (2)  an  index  of  physical  production,  meas- 
uring the  same  output  primarily  in  such  physical  units 
as  pounds,  bushels,  and  yards.  Somewhat  akin  to  the 
latter  is  an  index  of  the  physical  volume  of  trade, 
which  attempts  to  measure  the  total  number  of  physical 
units  traded  in  a  given  period.  Because  of  the  sea- 
sonal nature  of  a  large  part  of  industry,  indexes  of 
production  are  generally  based  on  the  year  as  the  unit 
of  time.^ 

Indexes  of  value  production  do  not  call  for  an  ex- 
tended discussion,  since  they  are  merely  inventories  of 
representative  annual  output  at  the  average  current 
prices.     The  process  of  finding  them  may  be  illus- 

*  One  of  the  most  practical  uses  of  production  index  numbers,  how- 
ever, requires  monthly  or  weekly  data,  corrected  for  seasonal  variations. 
These  data  are  beginning  to  be  used  in  connection  with  business 
barometrics,  as  will  be  noted  in  the  next  chapter. 

74 


QUANTITY  INDEXES  AND  THEIR  USES     75 

trated  by  multiplying  each  year's  output  of  wheat, 
corn,  cotton,  pig  iron,  and  copper,  as  shown  in  Table 
XIV,  by  their  respective  prices  as  shown  in  Table 
XIV- A.  The  resulting  annual  totals  may  be  taken  pro- 
visionally as  an  index  of  value  production  of  raw 
materials,  and  may  be  reduced  to  any  desired  base. 
Such  indexes  are  not,  however,  of  much  use  in  them- 
selves, except  as  they  may  be  employed  in  connec- 
tion with  the  computation  of  physical  production  and 
price  indexes. 

If  the  total  value  production  for  a  given  year,  as 
measured  by  an  adequate  index,  shows  an  increase  over 
the  preceding  year,  this  increase  may  evidently  be  at- 
tributed either  to  a  growing  volume  of  physical  pro- 
duction, or  to  rising  prices,  or  to  a  combination  of  both 
factors.  It  therefore  follows  that  suitable  indexes  of 
general  prices  and  of  physical  production,  multiplied 
across  year  by  year,  must  give  an  index  of  value  pro- 
duction. This  fundamental  principle  may  be  expressed 
by  the  formula :  ^ 

PnQn  =  Vn 

in  which 

Pn  =  the  price  index  for  a  given  year 

Qn  =  the  physical  production  index  for  the  same 
year 

Vn  =  the  value  production  index  for  the  same  year 
The  formula  may  also  be  written : 

Vn  Vn 

Pn  =  7^,  and  Qn  =  p- 

vjn  Xn 

An  Index  of  Physical  Production.     The  statistical 

*  In  applying  this  formula,  it  is  preferable  that  the  two  given 
indexes  should  be  reduced  to  the  same  base,  but  this  is  not  mathe- 
matically essential.  It  is  not  the  absolute  value  of  the  derived  index 
numbers  that  is  important,  but  only  their  ratios  to  each  other. 


TABLE  XIV » 
PRODUCTION  OF  SPECIFIED  COMMODITIES,  U.  S.,  1870-1920 


Yeae 

Wheat 

Corn 

Cotton 

Pig  Iron 

Copper 
(mil- 
lions op 

POUNDS) 

(millions 

(millions 

(millions 

(millions 

OP  bushels) 

OF  bushels" 

OF  bales) 

OF  tons) 

1870 

236 

1,094 

4.352 

1.665 

28 

1871 

231 

992 

2.974 

1.707 

29 

1872 

250 

1,093 

3.931 

2.549 

28 

1873 

281 

932 

4.170 

2.561 

35 

1874 

308 

850 

3.833 

2.401 

39 

1875 

292 

1,321 

4.632 

2.024 

40 

1876 

289 

1,284 

4.474 

1.869 

43 

1877 

364 

1,343 

4.774 

2.067 

47 

1878 

420 

1,388 

5.074 

2.301 

48 

1879 

449 

1,548 

5.755 

2.742 

52 

1880 

499 

1,717 

6.606 

3.835 

60 

1881 

383 

1,195 

5.456 

4.144 

72 

1882 

504 

1,617 

6.950 

4.623 

91 

1883 

421 

1,551 

5.713 

4.596 

116 

1884 

513 

1,796 

5.6S2 

4.098 

145 

1885 

357 

1,936 

6.576 

4.045 

166 

1886 

457 

1,665 

6.505 

5.683 

158 

1887 

456 

1,456 

7.047 

6.417 

181 

1888 

416 

1,988 

6.938 

6.490 

226 

1889 

491 

2,113 

7.473 

7.604 

227 

1890 

402 

1,490 

8.653 

9.203 

260 

1891 

612 

2,060 

9.035 

8.280 

284 

1892 

516 

1,628 

6.700 

9.157 

345 

1893 

396 

1,619 

7.493 

7.125 

329 

1894 

460 

1,213 

9.901 

6.658 

354 

1895 

467 

2,151 

7.161 

9.446 

381 

1896 

428 

2,284 

8.533 

8.623 

460 

1897 

530 

1,903 

10.898 

9.653 

494 

1898 

675 

1,924 

11.189 

11.774 

527 

1899 

5_47 

2,078 

9.393 

13.621 

569 

'1900  . 
1901 

522 

2n^05 

10.102 

13.789 

606 

748 

1,523 

8.583 

16.878 

602 

1902 

670 

2,524 

10.588 

17.821 

660 

1903 

638 

2,244 

9.820 

18.009 

698 

1904 

552 

2,467 

13,451 

16.497 

813 

1905 

693 

2,708 

10.495 

22.992 

889 

1906 

735 

2,927 

12.983 

25.307 

918 

1907 

634 

2,592 

11.058 

25.781 

869 

1908 

665 

2,669 

13.086 

15.936 

943 

1909 

737 

2,772 

10.073 

25.795 

1,093 

1910 

635 

2,886 

11.568 

27.304 

1,080 

1911 

621 

2,531 

15.553 

23.650 

1,097 

1912 

730 

3,125 

13.489 

29.727 

1,243 

1913 

763 

2,447 

13.983 

30.966 

1,224 

1914 

891 

2,673 

15.906 

23.332 

1,150 

1915 

1,026 

2,995 

11.068 

29.916 

1,388 

1916 

636 

2,567 

11.364 

39.435 

1,928 

1917 

637 

3,065 

11.302 

38.621 

.  1,890 

1918 

921 

2,503 

12.041 

39.055 

1,994 

1919 

941 

2,917 

11.421 

31.015 

1,289 

1920 

787 

3,232 

13.366 

36.415 

1,345 

^This  table  and  the  one  following  are  taken  with  some  modifications 
from  Babson  's  Business  Barometers,  by  permission  of  the  author. 

7G 


TABLE  XIV— A 

AVERAGE  PRICES  OF  SPECIFIED   COMMODITIES,  U    S 

(EASTERN  MARKETS),   1870-1920  ' 


TEAR 

WHEAT 

CORN 

COTTON 

PIG  IRON 

COPPER 

PER  BU. 

PER  BU. 

PER  BALE 

PER  TON 

PER  LB. 

1870 

1.30 

1.02 

119.50 

33.23 

0.211 

1871 

1.60 

.77 

84.50 

35.08 

.241 

1872 

1.62 

.70 

110.50 

48.94 

.355 

1873 

1.76 

.63 

100.50 

42.79 

,280 

1874 

1.39 

.86 

89.50 

30.19 

,220 

1875 

1.33 

.84 

77.00 

25.53 

,226 

1876 

1.35 

.628 

64.50 

20.75 

.210 

1877 

1.63 

.593 

59. 

19.25 

,190 

1878 

1.24 

,535 

56. 

17.05 

,165 

1879 

1.24 

.47 

54. 

22.82 

.186 

1880 

1.30 

.55 

57.50 

29.86 

.214 

1881 

1.30 

.62 

60. 

22.54 

.181 

1882 

1.32 

.77 

57.50 

23.20 

.191 

1883 

1.17 

.64 

59. 

19.62 

.165 

1884 

1.00 

.615 

54.50 

16.80 

.110 

1885 

.94 

.51 

52. 

15.20 

.108 

1886 

.888 

.52 

46. 

16.77 

.110 

1887 

.88 

.488 

51. 

20.05 

.138 

1888 

.94 

.593 

50. 

16.82 

,167 

1889 

.91 

.438 

53. 

14.35 

,134 

1890 

.92 

.485 

55. 

15.10 

,156 

1891 

1.05 

.675 

43. 

13.78 

.127 

1892 

.908 

.54 

38.50 

12.74 

.115 

1893 

.739 

.499 

42.50 

11.42 

.107 

1894 

.611 

.509 

34.50 

9.93 

.095 

1895 

.669 

.477 

37. 

10.86 

.105 

1896 

.781 

.340 

39.50 

10.29 

.109 

1897 

.954 

.319 

35. 

9.42 

.113 

1898 

.952 

.376 

29.50 

9.46 

.120 

1899 

.794 

.413 

34. 

16.58 

.177 

1900 

.804 

.453 

46. 

17.04 

.166 

1901 

.803 

.567 

43.50 

13.61 

.161 

1902 

.836 

.684 

45. 

20.00 

.116 

1903 

.853 

.572 

55.50 

17.08 

,132 

1904 

1.107 

.594 

58.50 

12.73 

.128 

1905 

1.028 

.593 

49. 

15.57 

.156 

1906 

.865 

.560 

57.50 

16.70 

,193 

1907 

.963 

.640 

60.50 

23,10 

,200 

1908 

1.049 

.786 

53. 

15.54 

,132 

1909 

1.263 

.767 

63. 

16.12 

,131 

1910 

1.118 

.668 

75.50 

15.16 

,129 

1911 

.963 

.711 

65. 

13.67 

.125 

1912 

1.091 

.711 

57.50 

14,93 

,164 

1913 

1.041 

.711 

64. 

14.90 

.155 

1914 

1.094 

.793 

55.50 

13.41 

,133 

1915 

1.291 

.837 

50.50 

13.58 

.174 

1916 

1.468 

.929 

72. 

18.67 

.272 

1917 

2.346 

1.776 

117.50 

40.07 

.272 

1918 

2.31 

1.840 

158.50 

36.52 

,247 

1919 

2.34 

1.771 

161.50 

32.16 

.192 

1920 

2.65 

1.669 

173. 

44.03 

.175 

77 


78  INTRODUCTION  TO  ECONOMIC  STATISTICS 

process  of  developing  an  index  of  physical  production 
may  be  illustrated  by  the  use  of  the  limited  data  shown 
in  Table  XIV.  In  aggregating  bushels,  bales,  tons, 
and  pounds  for  any  given  year,  it  will  be  hardly  admis- 
sible to  add  the  units  as  tabulated.  "While  the  numbers 
might  be  taken  abstractly  and  thus  combined  into  an 
index,  yet  such  a  process  would  give  as  much  weight 
to  a  bushel  or  a  pound  as  to  a  bale  or  a  ton.  It  is  true 
that  we  might  here  reduce  all  our  units  to  pounds,  but 
even  if  this  were  done  a  pound  of  copper  should  be 
stressed  more  than  a  pound  of  corn  or  of  iron,  because 
of  its  greater  importance  in  the  markets.  The  simplest 
way  to  give  each  item  of  production  its  proper  place  in 
the  total  will  be  to  remeasure  it  in  terms  of  a  standard 
value.  That  is,  we  may  take  as  the  physical  unit  the 
amount  of  each  commodity  that  can  be  bought  for  a 
dollar  at  a  standard  price.  The  number  of  such  units 
for  each  year  may  then  be  summated  as  an  index  of 
physical  production.^  Expressed  algebraically,  the 
process  is : 

Qn    =  Sp^qn 

In  so  far  as  the  complete  index  is  concerned,  this  is 
equivalent  to  averaging  the  quantities  as  originally 
tabulated,  weighting  them  for  standard  price  (pm). 

A  Standard  Price.  The  term  '  *  standard  price ' '  has 
been  used  to  imply  a  price  which  may  be  regarded  as 
representative  of  a  given  commodity  for  the  whole  in- 

*  It  can  be  shown,  as  follows,  that  the  product  of  price  and  quantity 
can  properly  be  taken  as  physical  units: 
Let  p  =  price  per  pound  in  dollars 
and  n  =  number  of  pounds  in  a  given  output. 
Then   1/p  =   number   of   pounds   purchasable   for   one   dollar    (the  new 

physical  unit), 
and  n  -i-  1/p  =  np,  the  number  of  now  physical  units  in  the  output. 


QUANTITY  INDEXES  AND  THEIR  USES     79 

terval  under  consideration.  Since  the  standard  price 
is  to  be  used  virtually  as  a  weight,  it  need  not  be  deter- 
mined with  very  great  accuracy.  But  from  the  point 
of  view  of  the  'Hheory  of  errors,"  it  should  be  the 
average  price  during  the  whole  interval  of  time  which 
is  covered  by  the  series  of  quantity  indexes  dependent 
upon  it.  It  follows,  therefore,  that  in  periods  of  rap- 
idly changing  prices,  somewhat  different  comparative 
results  may  be  obtained  by  varying  the  interval  of 
time  over  which  the  comparisons  are  made.  This  may 
seem  anomalous,  but  it  arises  inevitably  from  the  fact 
that  the  concept  of  aggregate  quantity  requires  a  unit 
dependent  upon  value ;  namely,  a  dollar's  worth.  Since 
value  is  unstable,  the  unit  of  quantity  is  also  unstable. 
In  practice,  however,  this  instability  is  usually  insig- 
nificant. 

The  application  of  the  foregoing  principles  to  the 
data  of  Table  XIV  requires  the  finding  of  a  standard 
price  for  each  of  the  commodities  tabulated.  As  a 
base  for  this  price,  a  period  of  about  twenty  years 
prior  to  the  war  has  been  chosen,  the  purpose  being  to 
exclude  war  prices  as  extreme,  and  to  emphasize  the 
later  rather  than  the  earlier  part  of  the  half  century 
studied.^  The  averages  have  been  taken  approxi- 
mately, and  have  been  slightly  modified  by  the  use  of 
data  not  here  cited.    They  are  as  follows : 

^  Because  of  the  shifting  of  relative  prices,  it  might  be  theoretically 
preferable  to  subdivide  the  half  century  under  consideration  into  at 
least  three  different  periods  (e.  g.,  1870-1896,  1896-1914,  and  1914- 
1920),  and  to  derive  different  sets  of  weights  for  use  in  each  period. 
The  series  of  indexes  could  be  brought  together  in  the  overlapping 
years  by  a  simple  adjustment  of  the  bases.  But  the  slight  gain  in 
accuracy  thus  made  would  not  be  worth  while  here,  since  in  any  case 
the  results  obtained  from  such  meager  data  cannot  be  considered  as 
anything  more  than  approximations. 


80  INTRODUCTION  TO  ECONOMIC  STATISTICS 

Wheat,  per  bushel $1.04 

Corn,  per  bushel 64 

Cotton,  per  bale 56.00 

Pig  Iron,  per  ton 16.00 

Copper,  per  pound 16 

In  applying  these  prices,  it  should  be  remembered  that 
they  are  in  effect  weights,  and  so  may  be  treated  as  a 
multiple  ratio.  They  may  therefore  be  changed  by 
division  into  the  more  convenient  form  13 :8 :700 :200 :2. 
Physical  Production  in  the  United  States.  After  the 
standard  prices  have  been  decided  upon,  each  annual 
item  of  production  is  multiplied  by  its  appropriate 
weight.  Two  sub-totals  are  taken  for  each  year,  one 
for  crops  and  one  for  minerals.  These  sub-totals  con- 
stitute two  provisional  index  series.  The  completed 
indexes  are  reduced  to  a  base  of  1913.  A  further  step 
may  be  taken  by  noting  the  fact,  as  shown  in  various 
statistical  studies,  that  an  index  of  mineral  production 
runs  very  close  to  an  index  of  manufactures.  By  suit- 
able weighting,  the  indexes  of  crops  and  iitinerals  may 
therefore  be  combined  so  as  to  include^  in  effect,  an 
estimate  for  the  value  added  by  manufacturing.  By 
a  comparison  of  aggregate  values  and  a  little  experi- 
mentation, the  requisite  weights  may  be  placed  at  six 
for  crops  and  four  for  minerals,  manufactures  being 
theoretically  included  in  the  latter.  This  process  of 
weighting  and  combining  is  admittedly  very  crude, 
and  would  not  be  expected  ordinarily  to  give  more 
than  a  rough  approximation  to  a  comprehensive  index. 
But  it  happens  in  this  case  that  the  five  commodities 
on  which  the  work  is  based  are  very  dependable  and 
typical  as  far  as  production  is  concerned.    The  index 


TABLE  XV 

INDEXES   OF  PHYSICAL   PEODUCTION,   UNITED   STATES, 

1870-1920 


YEAR 

CROPS 

MINERALS 

GEN  ER  All 

AGGREGATE 

PER  CAPITA 

1870 

38 

5 

25 

61 

1871 

33 

5 

22 

53 

1872 

38 

7 

25 

60 

1873 

36 

7 

24 

56 

1874 

34 

6 

23 

52 

1875 

45 

6 

29 

64 

1876 

44 

5 

28 

60 

1877 

48 

6 

31 

65 

1878 

51 

6 

33 

67 

1879 

57 

8 

37 

73 

1880 

63 

10 

42 

81 

1881 

47 

11 

33 

61 

1882 

62 

13 

42 

78 

1883 

56 

13 

39 

69 

1884 

64 

13 

43 

76 

1885 

63 

13 

43 

74 

1886 

61 

17 

43 

72 

1887 

57 

19 

42 

69 

1888 

67 

20 

48 

77 

1889 

73 

23 

53 

83 

1890 

59 

27 

46 

71 

1891 

78 

26 

57 

86 

1892 

62 

29 

49 

72 

1893 

59 

24 

45 

66 

1894 

58 

24 

44 

63 

1895 

72 

31 

55 

77 

1896 

76 

31 

58 

79 

1897 

76 

34 

59 

79 

1898 

81 

39 

65 

85 

1899 

77 

45 

64 

83 

1900 

78 

46 

65 

83 

1901 

71 

51 

63 

78 

1902 

92 

57 

78 

95 

1903 

84 

58 

74 

88 

1904 

92 

57 

78 

91 

1905 

97 

74 

88 

100 

1906 

107 

80 

96 

108 

1907 

93 

80 

88 

97 

1908 

100 

59 

83 

90 

1909 

99 

85 

93 

99 

1910 

100 

88 

96 

100 

1911 

100 

80 

92 

95 

1912 

112 

98 

106 

108 

1913 

100 

100 

100 

100 

1914 

112 

81 

100 

98 

1915 

115 

101 

109 

106 

1916 

94 

136 

110 

106 

1917 

104 

133 

115 

109 

1918 

102 

137 

116 

108 

1919 

111 

102 

107 

99 

1920 

115 

110 

113     1     103 

81 


82  INTRODUCTION  TO  ECONOMIC  STATISTICS 

obtained  from  them  in  fact  cheeks  remarkably  well 
with  Stewart's  (1890-1920)  and  Day's  (1899-1920)  in- 
dexes of  physical  production.  The  results,  put  into 
index  form,  are  sho^vn  in  Table  XV. 

Price  Indexes.  The  theory  of  price  indexes  ^  may  be 
advantageously  reviewed  in  the  light  of  the  principles 
discussed  in  this  chapter.  It  will  be  apparent  that  a 
price  index,  to  be  theoretically  precise,  must  conform 
to  the  equation, 

Since  this  equation  is  valid  whether  the  indexes  sub- 
stituted in  it  have  been  reduced  to  a  common  base  or 
expressed  merely  in  dollars  and  ''dollar's  worths,"  it 
may  be  written,  by  the  use  of  formulas  previously  con- 
sidered : 

p  _  ^Pnq° 

X  n  —  "^T  ' 

^Pmqn 

The  index  numbers  obtained  by  the  direct  use  of  this 
formula  will  have  as  their  base  approximately  an  aver- 
age index,  as  determined  by  the  use  of  average  prices 
in  the  denominator.  If,  however,  the  indexes  of  value 
and  quantity  have  first  been  reduced  to  a  given  base, 
then  the  price  indexes  will  have  that  base. 

*  The  theories  of  price  and  quantity  indexes  discussed  in  this  book 
involve  the  use  of  the  same  list  of  items  from  one  date  to  another.  If 
a  change  in  the  number  or  character  of  the  items  is  to  be  made,  as  the 
change  from  22  to  43  items  in  the  Bureau  of  Labor  Statistics'  index 
of  retail  food  prices,  it  is  assumed  that  a  new  series  is  begun,  and  is 
connected  with  the  old  merely  by  an  adjustment  of  the  new  base  so 
as  to  make  the  indexes  at  the  two  overlapping  dates  agree.  The  difficult 
theoretical  problem  of  constructing  an  index  based  upon  continually 
shifting  lists  of  commodities  is  not  considered,  since  it  has  little  imme- 
diate practical  value.  Production  indexes,  however,  to  be  quite  accurate, 
ought  to  be  gradually  broadened  to  take  account  of  the  growing  diversi- 
fication of  industry.  Hence  data  such  as  freight  and  canal  tonnage, 
which  reflect  this  increasing  diversification,  are  useful  in  measuring  pro- 
duction. 


QUANTITY  INDEXES  AND  THEIR  USES     83 

For  the  purpose  of  making  a  comparison  with  the 
proportional  expenditure  method,  the  formula  may 
be  modified  by  twice  inserting  the  factor  pm  in  such  a 
way  that  it  will  cancel  from  each  term  of  the  numerator 
summation,  as  follows : 

vP°    v/ 

■p  _      P'" 

J-  n  -^ 

^Pmqn 

The  formula  as  thus  written  indicates  relatives  based 
upon  standard  prices,  and  an  averaging  of  these  rela- 
tives with  weights  consisting  of  contemporary  physical 
quantities  measured  in  units  of  a  dollar's  worth  at 
standard  prices.  Briefly  stated,  the  formula  means 
relatives  weighted  for  "dollar's  worths."  Of  course 
in  actual  work  the  simpler  form  previously  stated 
^ould  be  used;  the  latter  form  is  given  merely  for 
comparison  and  interpretation.  Since  this  method  of 
finding  a  price  index  involves  a  standardization  of 
quantity  units  throughout  a  given  period,  it  may  be 
called  the  method  of  standard  quantities. 

An  Approximate  Method.  The  averaging  of  prices  by 
the  use  of  quantity  weights  suggests  an  approximate 
method  for  finding  price  indexes  which  may  here  be 
stated  by  way  of  further  comparison.  This  method 
uses  actual  rather  than  relative  prices,  and  is  weighted 
by  the  use  of  the  average  physical  quantities  (qm) 
produced,  consumed,  or  traded  during  the  period  of 
time  in  question.  It  is  analogous  to  the  method  used 
in  finding  quantity  indexes.    Its  formula  is : 

Pn  —  Sp^qm 
The  series  of  indexes  so  derived  may  be  reduced  to  any 
desired  base.    The  formula  is  a  convenient  one,  but 


84  INTRODUCTION  TO  ECONOMIC  STATISTICS 

it  is  not  theoretically  valid,  and  therefore  will  not  give 
very  dependable  results.  It  fails  to  meet  the  test  im- 
plied in  the  formula : 

PnQn  =  Vn 

Theoretical  Difficulties.  Certain  theoretical  difficul- 
ties encountered  in  developing  a  method  of  finding  a 
price  index  may  be  revealed  by  approaching  the  prob- 
lem from  another  angle.  Suppose,  first,  that  we  had  to 
deal  merely  with  one  commodity  having  a  value  for  a 
given  year  of  four  dollars,  and  for  the  succeeding 
year  of  five  dollars.  If  the  first  year  is  taken  as  a 
base,  the  price  index  for  the  second  year  is  125.  If 
the  second  year  is  taken  as  a  base,  the  price  index  for 
the  first  year  is  80.  These  two  results  are  consistent, 
as  may  be  shown  by  the  fact  that  80%  times  125%  is 
unity,  or  by  the  fact  that  the  reciprocal  of  the  index  80 
is  the  index  125. 

Let  us  now  consider  an  analogous  problem  involving 
two  commodities,  with  quantities  and  prices  stated,  as 
follows : 

FIRST  YEAR 

(q)  (P)       (v) 

Commodity  A,  10  lbs.  @  $4  =  $40 
Commodity  B,     3  bu.   @    7  =    21 

Value  index  61 

SECOND  YEAR 

(q)  (P)       (v) 

Commodity  A,  12  lbs.  @  $5  =  $60 
Commodity  B,     2  bu.    @    6  =     12 

Value  index  72 


QUANTITY  INDEXES  AND  THEIR  USES     85 

Instead  of  making  use  of  average  prices  as  standards 
in  computing  quantities,  we  shall  follow  a  common 
usage  and  take  the  prices  of  the  base  year.  Considering 
the  first  year  as  the  base,  and  its  prices  as  the  stand- 
ards for  finding  quantity  units  (dollar's  worths),  we 
have  61  as  both  the  value  and  the  quantity  index.  The 
price  index,  V  -f-  Q,  is  therefore  100,  as  it  should  be  in 
the  base  year.  In  the  second  year  V  =  72  and  Q  =  62, 
the  latter  result  being  obtained  by  summating  the 
quantities  of  the  second  year  at  the  standard  prices 
of  the  preceding  year.  The  price  index  for  the  second 
year  is  therefore  72%  -^-62%,  or  116%,  as  compared 
with  100%  for  the  first  year. 

Reversing  the  computation,  we  now  take  the  second 
year  as  the  base  year,  and  its  prices  as  the  standard 
prices.  The  price  index  for  the  second  year  will  now 
come  to  100,  since  the  value  index  is  identical  with  the 
quantity  index.  For  the  first  year  the  value  index  will 
be  61  as  before;  and  the  quantity  index,  obtained  by 
summating  the  quantities  at  the  standard  prices  of  the 
second  year,  will  be  68.  The  price  index  for  the  first 
year  is  therefore  61%  -^68%,  or  90%. 

Are  our  two  results,  obtained  from  different  bases, 
consistent  with  each  other,  as  they  were  when  only 
one  price  index  was  used"?  If  so,  the  product  of  the 
two  indexes,  116%  and  90%,  should  be  unity.  But 
their  product  is  actually  104%).  The  other  test  is  that 
the  reciprocal  of  the  index  90%  should  equal  the  index 
116%.    But  actually  the  reciprocal  of  90%  is  111%. 

It  is  obvious  that  the  lack  of  consistency  arises  from 
the  failure  to  standardize  quantity  units  by  the  use  of 
a  constant  price.     If,  now,  we  apply  the  method  of 


86  INTRODUCTION  TO  ECONOMIC  STATISTICS 

standard  quantities,  we  obtain  the  following  results  for 
the  two  years,  respectively : 

P.  =  |=94  P.  =  r7=107 

Considering  the  first  year  as  a  base,  we  obtain  an  in- 
dex of  114  for  the  second  year.  If  the  second  year  is 
taken  as  the  base,  a  consistent  result  will  necessarily 
be  obtained.  It  will  be  seen  that  the  index  for  the 
second  year  thus  obtained  is  nearly  midway  between 
the  two  indexes,  116  and  111,  obtained  before. 

Fisher's  Index.  Thus  the  standard  quantity  method 
may  be  used  to  avoid  the  inconsistency  encountered 
when  different  standard  prices  are  used  in  measuring 
quantities.  But  Professor  Fisher  has  proposed  an- 
other solution  which  he  regards  as  the  theoretical  ideal. 
His  method  involves  the  finding  of  the  geometrical 
average  of  the  two  inconsistent  results,  as  seen  in  the 
two  indexes,  116  and  111.  This  method  will  here  give 
a  result  of  114,  which  is  equal  to  that  obtained  by 
the  method  of  standard  quantities,  though  as  a  rule 
the  results  are  only  approximately  equal.  The  latter 
method  has  a  distinct  advantage  in  that  it  will  give  a 
consistent  series  of  index  numbers  for  a  required  inter- 
val of  time,  and  the  series  may  be  reduced  to  any  de- 
sired base.  The  weight  of  authority,  however,  supports 
Professor  Fisher's  solution  as  being  theoretically  the 
best.  His  formula,  which  is  somewhat  too  complex  for 
ordinary  use,  is  as  follows : 


2poqn         Spoqo 
in  which  the  subscripts  o  and  n  indicate  a  base  year 
and  a  relative  year,  respectively.    The  development  of 


QUANTITY  INDEXES  AND  THEIR  USES     87 

the  formula  is  sufficiently  explained  in  the  foregoing 
discussion.  The  corresponding  formula  for  quantities 
(Qn)  may  be  written  by  interchanging  each  p  and  q 
in  the  price  formula. 

It  will  hardly  be  profitable  to  attempt  to  follow  any 
further  the  theoretical  problems  connected  with  the 
compiling  of  price  and  quantity  indexes.  It  must  be 
emphasized  that  at  present  the  more  significant  prac- 
tical problems  relate  to  the  securing  of  adequate  and 
reliable  data  rather  than  to  the  precision  of  the  meth- 
ods used  in  their  elaboration.  The  choice  of  method, 
as  a  rule,  will  not  involve  the  danger  of  serious  error ; 
while  work  may  be  quite  invalidated  by  deficiencies 
and  inaccuracies  in  the  data.  An  understanding  of  the 
theory  of  the  subject  will  nevertheless  be  found  to  have 
a  practical  value  in  the  planning  of  work  and  in  the 
evaluation  of  results. 

Quantity  Theory  Indexes.  An  interesting  applica- 
tion of  price  and  production  indexes  to  the  measure- 
ment of  general  business  changes  may  be  made  on  the 
basis  of  the  quantity  theory  of  money.  This  theory 
in  its  simplest  form  states  that  prices  (P)  change 
directly  as  the  circulating  medium  (M)  and  its  rate  of 
circulation  (R),  and  inversely  as  the  number  of  phys- 
ical units  traded  (N).  This  statement  is  expressed 
algebraically  as  the  following  equation  of  exchange : 

MR 
^~      N     • 
The  equation  may  be  elaborated  by  subdividing  the 
circulating  medium  into  its  various  elements,  as  money 
and  credit,  and  dealing  with  each  on  the  basis  of  its 
own  average  rate  of  circulation.    But  as  it  is  difficult  to 


88  INTRODUCTION  TO  ECONOMIC  STATISTICS 

obtain  accurate  data  bearing  upon  circulation  rates, 
such  elaboration  will  not  be  attempted  here.  It  should 
also  be  noted  concerning  the  quantity  theory  that  there 
has  been  much  argument  as  to  its  validity.  But  the 
arguments  have  not  been  concerned  primarily  with  the 
statistical  aspects  of  the  subject,  to  which  little  objec- 
tion has  been  made.  They  have  dealt,  rather,  with  the 
origin  of  changes  in  the  equilibrium  expressed  by  the 
equation. 

In  order  to  make  a  certain  provisional  use  of  the 
formula,  further  data  must  be  adduced.  Values  for 
the  term  M  may  be  obtained  from  a  study  made  by 
Professor  E.  W.  Kemmerer,  entitled.  High  Prices 
and  Deflation,  which  gives  estimates  of  the  money 
and  bank  credit  in  circulation  during  the  war  period. 
Brought  down  to  1920,  these  estimates  are  as  follows : 

Circulation 
(Millions  of  dollars) 
Year  Money  Deposits 

1913 3,390  12,678 

1914 3,505  13,430 

1915 3,682  14,411 

1916 4,159  17,840 

1917 4,914  21,273 

1918 5,579  23,771 

1919 5,793  27,928 

1920 6,060  30,300 

In  combining  money  and  deposits  into  a  single  index 
of  circulating  medium,  it  is  necessary  to  multiply  de- 
posits by  two  because  they  circulate,  in  the  form  of 
checks,  about  twice  as  fast  as  money.    The  numbers 


QUANTITY  INDEXES  AND  THEIR  USES     89 

obtained  by  addition  may  then  be  reduced  to  a  1913 
base.  For  the  value  of  iV,  index  numbers  of  physical 
production  will  here  be  substituted.  It  is  assumed  that 
this  may  be  done  because,  as  a  rule,  the  volume  of 
trade  parallels  the  volume  of  production.  For  the 
value  of  P,  the  Bureau  of  Labor  Statistics*  index  of 
wholesale  prices  will  be  used. 

An  index  of  the  rate  of  currency  circulation  may 
now  be  readily  obtained  algebraically  by  the  use  of  the 
equation  of  exchange  expressed  in  the  form 

PN 

^-  "ir- 

The  resulting  index  will  not,  of  course,  register  at  all 
accurately  the  percentage  changes  in  the  rate  of  cur- 
rency circulation,  owing  to  the  fact  that  changes  in  the 
rate  of  circulation  of  goods  have  been  omitted  by  the 
substitution  of  volume  of  production  for  volume  of 
trade.  But  it  should  show  changes  in  the  activity  of 
business  by  registering  the  increases  or  decreases  of 
monetary  circulation  relative  to  corresponding  changes 
in  the  circulation  of  goods.  That  is,  it  should  indicate 
the  relative  activity  of  the  consumption  market. 
Speculative  flurries,  which  increase  the  circulation  of 
both  currency  and  securities,  will  therefore  not  regis- 
ter at  all.  That  the  index  does,  in  fact,  show  changes 
in  the  activity  of  business  from  year  to  year  with  some 
degree  of  certainty,  is  apparent  from  the  figures.  The 
depression  of  1914,  the  war  activity  of  1917,  the  tem- 
porary slump  of  1919,  and  the  boom  culminating  in 
1920  are  all  indicated. 

The  indexes  entering  into  the  equation  of  exchange 
as  just  discussed  are  given  below.     Taken  together, 


90  INTRODUCTION  TO  ECONOMIC  STATISTICS 

they  present  an  epitome  of  the  general  movement  of 
business.  Prices  vary  directly  with  purchasing  power, 
MR  (demand),  and  inversely  with  production,  N  (sup- 
ply). It  should  be  observed  that  the  values  of  R  and  N 
here  used  are  mere  approximations.  They  are  prob- 
ably too  high  in  1914  and  1915,  and  too  low  in  1916. 


Year 

(P) 

(M) 

(E) 

(N) 

1913 

100 

100 

100 

100 

1914 

100 

106 

94 

100 

1915 

101 

113 

97 

109 

1916 

124 

139 

98 

110 

1917 

176 

165 

123 

115 

1918 

196 

185 

123 

116 

1919 

212 

214 

106 

107 

1920 

243 

232 

118 

113 

A  complete  statistical  verification  of  the  equation  of 
exchange  would  necessitate  ampler  data  and  more  ex- 
tended computations.  The  point  of  greatest  difficulty 
would  be  found  to  be  the  construction  of  an  index  of 
the  physical  volume  of  trade  (N).  Theoretically  this 
should  include  all  wealth  and  services  traded  in  a  given 
period  of  time,  measured  in  physical  units  of  ''dollar's 
worths"  at  average  prices  for  the  whole  interval  over 
which  the  computation  extends.  Since  MR  is  equal  to 
the  value  of  the  same  quantities  at  current  prices,  the 
equation  of  exchange  reduces  to  the  formula  for  find- 
ing price  index  numbers  by  the  standard  quantities 
method ;  thus : 

°  ~     N       ~   Spmqn 

Of  course  in  actual  work  a  process  of  sampling  must 
necessarily  be  employed,  and  various  methods  for  ag- 


QUANTITY  INDEXES  AND  THEIR  USES     91 

gregating  quantities  may  be  resorted  to.  The  price 
index  thus  obtained  would  theoretically  vary  somewhat 
from  the  usual  type  of  index,  because  the  quantities  in- 
dicated in  the  formula  refer  to  trade  rather  than  to 
production  or  consumption.  Hence  goods  in  which 
there  is  unusually  active  speculation  will  in  effect  be 
weighted  heavily. 

For  discussions  of  the  quantity  theory  and  statis- 
tical verifications  of  the  equation  of  exchange  the  stu- 
dent may  profitably  consult  Kemmerer,  Money  and 
Credit  Instruments  in  their  Relation  to  General  Prices; 
Fisher,  The  Purchasing  Power  of  Money;  and  articles 
published  by  King  in  the  weekly  service  of  the 
Bankers'  Statistics  Corporation  (1920). 

Measurement  of  the  National  Income.  We  have  thus 
far  considered  production  indexes  principally  with  ref- 
erence to  their  use  in  connection  with  price  indexes. 
But  they  are  also  of  direct  significance  in  that  they 
indicate  changes  in  the  total  national  income.  By 
national  income  is  meant  the  sum  of  all  consumption 
values  in  economic  goods,  services,  and  property 
usances,  together  with  increases  in  stores  of  goods  and 
extensions  of  property.  This  conception  differs  slightly 
from  aggregate  income  as  privately  reckoned,  inas- 
much as  the  latter  includes  increases  in  capitalized 
values,  particularly  of  land. 

While  production  indexes  show  changes  in  the  na- 
tional income,  they  serve  merely  as  ratios  unless  sup- 
plemented by  estimates  of  the  total  income  for  one  or 
more  years.  Several  such  studies  have  been  attempted, 
but  the  most  complete  and  authoritative  study  is  one 
recently  made  by  the  National  Bureau  of  Economic 


92  INTRODUCTION  TO  ECONOMIC  STATISTICS 

Research.  This  Bureau  is  a  private  organization 
chartered  in  1920  for  the  purpose  of  conducting  statis- 
tical investigations  into  subjects  affecting  the  public 
welfare.  It  is  controlled  by  a  board  of  nineteen  di- 
rectors representing  various  points  of  view  and  in- 
terests. Its  staff  consists  of  Wesley  C.  Mitchell,  Will- 
ford  I.  King,  Frederick  R.  Macauley,  and  Oswald  W. 
Knauth.  This  organization  has  recently  issued  a  mon- 
ograph containing  estimates  of  the  national  income 
for  each  of  the  years  1909-1918,  inclusive.  The  mono- 
graph is  worthy  of  careful  study,  not  only  for  its  con- 
clusions, but  also  as  an  excellent  example  of  applied 
statistical  methods.  The  estimates  of  income  were 
made  independently  on  two  different  bases;  first,  by 
sources  of  production,  and  second  by  incomes  re- 
ceived. The  two  sets  of  results  agree  very  closely, 
the  maximum  difference  being  6.9%  in  1913.  They 
were  averaged  to  give  the  final  estimates,  which  were 
reduced  to  terms  of  1913  prices.  As  thus  reduced,  the 
indexes  were  as  follows,  the  income  for  1913  being 
34.4  billion  dollars : 


Year 

Index 

1909 

88 

1910 

94 

1911 

92 

1912 

97 

1913 

100 

1914 

96 

1915 

102 

1916 

118 

1917 

119 

1918 

113 

The  Distribution  of  Income. 

The  National  Bureau 

of  Economic  Research  has  also  made  a  study  of  the 
individual   distribution   of  income   in   1918,  based  in 


QUANTITY  INDEXES  AND  THEIR  USES     93 

large  part  on  income  tax  returns.  The  total  income 
for  that  year  was  found  to  be  about  58  billion  dollars, 
and  the  number  of  personal  income  recipients  was 
37.6  million,  exclusive  of  men  in  active  service  in  re- 
spect to  both  items.^  The  average  income  was  $1,543 ; 
the  mode,  $957;  and  the  three  quartiles  were  $833, 
$1,140,  and  $1,574,  respectively.  The  distribution  is 
summarized  in  the  following  derived  tables : 

FEEQUENCY  DISTRIBUTION  OF  NATIONAL  INCOME  IN  PER- 
CENTAGES OF  CIVILIAN  INCOME  RECIPIENTS,  U.  S.,  1918 


Income  class 

Per  cent 

Under  $400 

2.84 

$  400-  800 

19.51 

800-1200 

32.16 

1200-1600 

21.53 

1600-2000 

9.88 

2000-2400 

4.64 

2400-2800 

2.63 

2800-3200 

1.61 

3200-3600 

1.07 

3600-4000 

.74 

4000  &  over 

3.39 

E  PERCENTAGES   OF 

CIVILIAN  INCOMl 

3  AND  OF  NATIONAL 

INCOME,  U.  S.,  19 

Recipients 

Aggregate 

of  incomes 

incomes 

10 

2.7 

20 

7.2 

30 

12.5 

40 

18.7 

50 

26.0 

60 

33.6 

70 

42.2 

80 

52.5 

90 

65.2 

99 

86.2 

100 

100.0 

RECIPI- 


^When  all  are  included,  the  average  income  becomes  $1490. 


94  IXTEODUCTIOX  TO  ECOXOMIC  STATISTICS 

The  first  of  these  tables  may  he  graphed  as  a  fre- 
quency curve,  omittiug  the  last  class  of  "$4,000  and 
over,"  which  if  accurately  represented  should  be  ex- 
tended to  include  incomes  of  several  miUious.^  The 
second  may  be  graphed  as  a  Loreuz  curve,  the  first 
column  being  represented  on  the  base  line,  and  the 
vertical  scale  being  drawn  equal  to  the  horizontal  scale. 
A  diagonal  drawn  upward  from  left  to  right — the  fine 
of  equal  distribution — will  serve  as  a  basis  of  compari- 
son. Or.  the  second  table  may  be  more  simply  graphed 
by  obtaining  by  subtraction  the  non-cumulative  per- 
centages of  aggregate  income,  and  representing  them 
as  successive  vertical  blocks  above  a  base  hue  on  which 
are  measured  the  successive  ten  per  cent  groups  of 
income  recipients. 

Pareto's  Law.  If  the  percentages  in  the  fre- 
quency distribution  are  summated  in  reverse  order, 
and  are  plotted  vertically  against  the  corresponding 
lower  class  limits  on  double-logarithmic  paper  (double 
cycle  on  each  scale),  the  so-called  Pareto's  law  of  in- 
come distribution  will  be  illustrated.    This  law  states 

^  If  the  frequency  distribution  of  incomes  is  graphed,  as  repre- 
sented, on  semi-logarithmic  paper,  with  the  number  of  incomes  (income 
recipients)  plotted  on  the  vertical  arithmetic  scale,  and  the  magnitude 
of  the  incomes  plotted  on  the  horizontal  logarithmic  scale,  a  somewhat 
regular  frequency  curve  is  formed.  This  indicates  a  type  of  distribu- 
tion which  is  normally  symmetrical  when  the  ratio  departures  from 
the  geometric  mean,  rather  than  the  differences  from  the  arithmetic 
mean,  are  taken  as  the  basis  for  measuring  the  dispersioiL  In  such  a 
distribution  the  median  is  therefore  approximately  the  geometric  mean 
of  the  first  and  third  quartiles.  instead  of  the  arithmetic  mean. 


S— 
2— 
1— 
0— 


I 


QUAXTITY  INDEXES  AXD  THEIR  USES    95 

that  the  cm-ve  of  income  distribution  above  the  mode, 
when  logarithmically  plotted,  approximates  a  straight 
line.  Much  the  same  result  may  also  be  obtained  by 
transferring  the  Lorenz  cuiTe  to  double-logarithmic 
paper ;  or  it  may  be  more  simply  evidenced  by  plotting 
the  original  frequency  curve  on  similar  paper.  Pa- 
reto's  law  merely  caUs  attention  to  the  type  of  fre- 
quency curve  normally  followed  by  income  data.  Thi^ 
is  a  curve  which  when  graphed  in  the  ordinary  manner 
is  seen  to  be  strongly  skewed  to  the  right — somewhat 
the  same  type  as  was  suggested  by  the  wage  data 
studied  in  the  first  two  chapters.  In  foiTuulating  the 
law  Pareto  thought  he  had  discovered  a  somewhat  in- 
flexible and  permanent  fact  of  economic  relationships. 
While  he  doubtless  over-emphasized  the  inflexibility  of 
the  law,  yet  he  nevertheless  pointed  out  an  interesting 
illustration  of  the  strong  tendency  to  statistical  regu- 
larity inherent  in  biological  and  social  phenomena- 
Income  in  Other  Countries.  The  National  Bureau 
of  Economic  Eesearch  has  also  made  estimates  of  the 
per  capita  income  in  several  countries  for  the  year 
1914.  These  estimates  are  based  upon  prior  studies 
made  by  a  well-known  EngHsh  authority.  Sir  Josiah 
Stamp.    The  Bureau's  data  give  the  following  ratios: 


United  States 

100% 

AustraUa 

79 

Tnited  Kingdom 

73 

Canada 

5S 

France 

55 

Germanv 

U 

Italy 

33 

Austria-Hungary 

30 

Spain 

16 

Japan 

9 

96   INTRODUCTION  TO  ECONOMIC  STATISTICS 

The  Distribution  of  Property.  A  problem  closely 
related  to  the  distribution  of  income  is  the  distribution 
of  properiy  ownership.  For  the  United  States  the  best 
known  investigation  of  the  subject  is  one  published  by 
Professor  W.  I.  King  in  1915.  The  total  wealth  for 
1910  was  estimated  to  have  been  about  200  billion  dol- 
lars, and  its  ownership  was  distributed  in  a  form  that 
is  suggested  by  the  curve  of  income.  The  richest  two 
per  cent  of  the  families  were  thought  to  own  over  fifty 
per  cent  of  the  property.  But  the  data  on  which  such 
estimates  rest  involve  many  uncertainties,  and  the 
meaning  of  the  results  is  further  obscured  by  lack  of 
knowledge  of  the  rate  at  which  fortunes  are  acquired 
and  dissipated. 

REFERENCES 

National  Bureau  of  Economic  Research,  Income  in  the  United 
States,  Volume  I. 

Day,  Edmund  E.,  **An  Index  of  the  Physical  Volume  of  Pro- 
duction," Review  of  Economic  Statistics,  September,  1920- 
January,  1921. 

Fisher,  Irving,  "The  Best  Form  of  Index  Numbers"  (and 
discussion),  Quarterly  Publications  of  the  American  Statis- 
tical Association,  March,  1921,  pp.  533-551. 

Fisher,  Irving,  The  Purchasing  Power  of  Money. 

Hoffman,  F.  L.,  "The  Economic  Progress  of  the  United  States 
During  the  Last  Seventy -five  Years,"  Quarterly  Publica- 
tions of  the  American  Statistical  Association,  December, 
1914,  pp.  294-318.  For  further  data  on  the  same  topic,  see 
Appendix  III. 

Ingalls,  W.  R.,  "Labor  the  Holder  of  the  Nation's  Wealth  and 
Income,"  The  Annalist,  September  13,  20,  and  27,  1920. 

King,  W.  I.,  Wealth  and  Income  of  the  People  of  the  United 
States. 

Meeker,  Royal,  "On  the  Best  Form  of  Index  Numbers," 
Quarterly  Publications  of  the  American  Statistical  Asso- 
ciation, Sept.,  1921,  pp.  909-915. 

Stewart,  Walker  W.,  "An  Index  Number  of  Production" 
(and  discussion),  American  Economic  Review,  March,  1921, 
pp.  57-81. 


QUANTITY  INDEXES  AND  THEIR  USES     97 

Walsh,  C.  M.,  The  Measurement  of  General  Exchange  Value. 
Working,  Holbrook,  ' '  What  is  to  be  the  Future  Price  Level  ? ' ' 
The  Annalist,  June  27,  1921,  p.  686. 

EXERCISES 

1.  From  Tables  XIV  and  XIV- A  (pages  76  and  77)  find  an 
index  of  value  production  (a)  for  crops  and  (b)  for  min- 
erals.   Reduce  each  index  to  a  1913  base. 

2.  Divide  each  item  in  the  index  of  value  production  of 
minerals  just  obtained,  by  the  corresponding  item  in 
the  index  of  physical  production  of  minerals  in  the 
United  States.    Explain  the  results. 

3.  Divide  each  item  in  the  index  of  value  production  of 
crops  by  the  corresponding  item  in  the  index  of  physical 
production  of  crops.     Explain  the  results. 

4.  Plot  on  semi-logarithmic  paper  the  index  obtained  in  the 
preceding  exercise,  together  with  the  index  of  whole- 
sale prices  in  the  United  States.  Explain  the  divergence 
of  the  two  trends. 

5.  ''The  Monthly  Review,"  issued  at  the  Federal  Reserve 
Bank  of  New  York,  gives  the  value  of  the  ten  leading 
crops  in  the  United  States  at  average  (standard)  prices 
for  recent  years  as  follows: 

Value 
(millions  of  ■ 
Year  dollars) 

1910  5,873 

1911  5,491 

1912  6,549 

1913  5,750 

1914  6,397 

1915  6,831 

1916  5,907 

1917  6,507 

1918  6,418 

1919  6,626 

1920  7,284 

1921  6,118 

From  these  data  derive  an  index  of  agricultural  produc- 
tion (base,  1913),  and  compare  it  graphically  with  the 
corresponding  index  computed  from  three  leading  crops 
(Table  XV,  page  81). 


98  INTRODUCTION  TO  ECONOMIC  STATISTICS 

6.  From  the  data  of  crops  and  prices  given  below,  find  index 
numbers  of  physical  production,  value  production,  and 
prices,  for  the  year  1914.  Express  each  index  in  terms 
of  1913  as  a  base.  (It  is  assumed  that  the  prices  of  1913 
represent  average  prices  for  a  certain  period.) 

Wheat  Corn  Cotton 

Year         Bu.       Price         Bu.        Price    Bales       Price 

1913  760        $1.05        2450      $0.72         14        $61.00 

1914  890  1.10        2670  .80        16  53.00 

7.  From  the  following  data  find  indexes  of  the  four  terms 
used  in  the  equation  of  exchange  (quantity  theory)  for 
the  year  1910,  taking  the  year  1900  as  the  base. 

Year:  1900        1910 

Price  index   80  100 

Money  in  circulation 2.2  3        (billions) 

Deposits  subject   to   check....     9.4        14.25         '' 
Physical  production    20  30  " 

8.  Plot  together  on  semi-logarithmic  paper  the  index  of 
wholesale  prices  in  the  United  States  for  1890  to  1920 
inclusive,  and  a  similar  index  (base,  1913)  of  per  capita 
circulation  derived  from  the  following  data : 

1890             $22.82  1905             $31.08 

23.45  32.32 

24.60  32.22 

24.06  34.72 

24.56  34.93 

1895                23.24  1910                34.33 

21.44  34.20 

22.92  34.34 

25.19  34.56 

25.62  34.35 

1900                26.93  1915                35.44 

27.98  39.29 

28.43  45.74 

29.42  50.81 

30.77  54.33 
1920                57.04 

9.  The  two  indexes  of  physical  production  in  the  United 
States  (1913-1920)  given  below,  are  adapted  (a)  from 
Day's  and   (b)    from  Stewart's  comprehensive  studies. 


QUANTITY  INDEXES  AND  THEIR  USES     99 

Using  data  for  prices  and  circulating  medium  cited  in  the 
discussion  of  the  quantity  theory  of  money,  derive  two 
indexes  for  R,  1913-1920,  and  compare  them  with  the 
corresponding  index  included  in  the  data  just  men- 
tioned. 

Year  (A)  (B) 


1913 

100 

100 

1914 

98 

100 

1915 

105 

111 

1916 

111 

116 

1917 

114 

123 

1918 

113 

124 

1919 

106 

119 

1920 

111 

122 

CHAPTER  V 
TRENDS  AND  CYCLES 

The  Nature  of  Trends.  The  interpretation  of  a  time 
series  of  index  numbers  usually  requires  the  determi- 
nation of  the  trend.  A  trend  is  a  derived  series  of  in- 
dex numbers  following  the  general  course  of  the  given 
items,  but  shortening  or  eliminating  the  fluctuations. 
Its  significance  is  best  grasped  by  means  of  a  graphic 
representation.  An  inspection  of  such  graphic  work 
will  show  that  trends  may  vary  from  straight  lines  on 
the  one  hand,  to  curves  almost  conforming  to  the 
given  items  on  the  other.  Whether  a  trend  is  dra^vn 
as  a  straight  or  an  irregular  line  depends  in  part  on 
the  nature  of  the  data  and  in  part  on  the  purpose  to  be 
served. 

The  Free-hand  Method.  A  good  draftsman  com- 
monly determines  trends  merely  by  charting  his  data 
and,  without  any  preliminary  computations,  drawing 
a  straight  or  curved  median  line  through  them.^  Such 
work  may  be  done  entirely  free-hand,  or  by  the  use 
of  irregular  curves  and  other  drafting  material,  but  in 
either  case  it  is  classed  as  a  free-hand  method.  For 
many  purposes  this  method  will  prove  satisfactory, 
and  should  be  practiced  by  the  student  until  it  can  be 
used  wdth  facility.     For  even  though  more  elaborate 

*  If  a  trend  based  upon  {reometric  means  is  desired,  the  data  may  be 
plotted  on  semi-logarithmic  paper. 

100 


TRENDS  AND  CYCLES 


101 


methods  are  to  be  applied  in  actual  work,  practice  in 
the  free-hand  drawing  of  trends  will  be  helpful.  It 
will  develop  an  appreciation  of  the  requirements  of  a 
given  problem,  without  which  the  best  of  mathematical 
methods  are  likely  to  be  misapplied. 

An  example  of  the  free-hand  method  of  drawing  a 
trend  is  given  in  the  graph  of  the  index  of  real  wages, 


IS70 


mo 


mo 


im 


1910 


mo 


FiQUBE  6.  Trends  and  cycles.  Upper  line,  index  of  real  wages  (hour 
rates)  in  the  United  States  (see  Table  X)  and  its  trend;  middle  line, 
index  of  per  capita  production  (see  Table  XV)  and  its  trend;  lower 
line,  cycles  of  wholesale  prices   (see  Figure  9). 

Repriiited,  with  permission^  from  Tlxe  Quarterly  Journal  of  tJie  Univer- 
sity of  North  Dakota,  January,  1922. 

in  Figure  6.  In  this  case  a  more  precise  method  would 
not  be  applicable  because  of  the  inaccuracies  caused 
by  the  substitution  of  wholesale  prices  for  the  cost  of 
living  in  the  computation.  As  was  previously  stated, 
there  is  reason  to  think  that  the  marked  rise  in  real 
wages  appearing  during  the  decade  1890-1900  is  an 
exaggeration.    The  trend  was  therefore  drawn  so  as  to 


102  INTRODUCTION  TO  ECONOMIC  STATISTICS 

discount  this  rise.  As  a  result  it  does  not  conform  to 
the  usual  rule  that  the  deviations  above  and  below  it 
should  balance. 

Method  of  Semi-Averages.  When  a  straight-line 
trend  is  to  be  drawn  through  a  fairly  long  series,  the 
free-hand  method  may  be  improved  upon  by  means  of 
a  simple  calculation.  The  average  of  the  series  may 
be  plotted  on  the  middle  ordinate,  and  the  straight- 
line  trend  drawn  by  inspection  through  this  point. 
The  plus  and  minus  deviations  must  then  necessarily 
balance.  Or,  better  still,  the  series  may  be  divided 
into  two  equal  parts  and  an  average  taken  for  each 
part.  These  averages  may  be  plotted  on  the  middle 
ordinate  of  each  half  series,  respectively,  and  the  trend 
drawn  through  the  two  points.  If  the  series  consists 
of  an  odd  number  of  items,  the  middle  item  and  unit 
of  time  may  be  divided  between  the  two  parts.  This 
method  of  semi-averages  is  the  one  that  was  used  in 
drawing  the  trend  of  per  capita  production  in  Figure 
6.  The  trend  thus  obtained  is,  in  a  long  series,  nearly 
identical  with  the  so-called  line  of  least  squares  to  be 
discussed  later. 

The  Moving  Avera,ge.  Among  the  trends  that  are 
found  by  mathematical  methods,  perhaps  the  best 
kno^vTi  is  the  moving  average.  The  process  of  com- 
puting the  moving  average  is  simple  in  principle,  but 
it  is  usually  tedious  in  practice  even  when  calculating 
machines  are  used.  By  this  method  the  position  of 
the  trend  at  any  given  period  in  the  series  is  found 
by  averaging  a  certain  number  of  items  centering  at 
that  period.  Just  how  many  items  should  be  included 
in  the  average  must  be  determined  by  the  nature  of 


TRENDS  AND  CYCLES  103 

the  data.  If  the  series  shows  a  cyclic  movement  of 
known  length,  then  by  taking  the  number  of  items 
covering  this  length  of  time,  the  cycles  will  be 
smoothed.  If  monthly  data  having  a  pronounced  sea- 
sonal swing  are  being  studied  through  an  interval  of 
several  years,  a  twelve-month  moving  average  is  ap- 
propriate.^ The  method  of  finding  the  moving  aver- 
age is  illustrated  in  the  following  table : 


Per  Capita 

Moving  A 

-veragei 

Year 

Production 

3-Year 

5.Yea 

1910 

100 

1911 

95 

101 

1912 

108 

101 

100 

1913 

100 

102 

101 

1914 

98 

101 

104 

1915 

106 

103 

104 

1916 

106 

107 

105 

1917 

109 

108 

106 

1918 

108 

105 

105 

1919 

99 

103 

1920 

103 

In  explanation  of  the  foregoing  table  it  may  be  said 
that  the  first  number  in  the  three-year  moving  average 
(101)  is  obtained  by  averaging  the  first,  second,  and 
third  items  (100,  95,  and  108).  The  second  number 
(101)  is  obtained  by  averaging  the  second,  third,  and 
fourth  items  (95,  108,  and  100).  The  results  are  writ-' 
ten  to  the  nearest  unit,  and  are  placed  opposite  the 
middle  one  of  the  three  items  averaged.  In  the  same 
way  the  succeeding  averages  are  derived.  The  five- 
year  moving  average  is   similarly  computed,  except 

*  The  twelve-month  moving  average  centers  between  the  sixth  and 
seventh  month  in  each  computation.  In  order  to  make  the  trend  thus 
obtained  fall  on  the  same  ordinates  as  the  original  items,  it  is  neces- 
sary to  adjust  it  by  deriving  from  it  a  two-month  moving  average.  The 
deviations  of  the  original  items  may  then  be  readily  obtained. 


104  INTRODUCTION'  TO  ECONOMIC  STATISTICS 

that  five  items  are  averaged  at  a  time.  In  practice 
the  work  may  be  somewhat  abridged,  after  the  total 
of  the  first  group  of  items  is  found,  by  deriving  the 
next  total  from  it.  This  may  be  done  by  adding  to 
the  first  total  the  difference  between  the  next  item 
about  to  be  included  and  the  one  about  to  be  dropped. 


WS^ 

A                   /n^ 

f06^ 

A       f^~\ 

m^ 

i 

1  \    r'4/        \\ 

/02^ 

J\::^'iy/                 \  / 

7 

'f\               v 

93^      \ 

/ 

^ 

96.        ^ 

4 

9^.     , 

V 

1 

1     1     1    1     1    1    1    1    1 

/P/0 

/P/S                        /9Z0 

Figure  7.  Moving  averages.  A,  Index  of  per  capita  production, 
1910-1920.  B,  Three-year  moving  average  of  same.  C,  Five-year  mov- 
ing average  of  same. 

Thus,  the  first  five  items  above  give  501  (average, 
100).  In  obtaining  the  total  of  the  second  to  the  sixth 
items,  inclusive,  the  sixth  item  (106)  will  be  added 
and  the  first  item  (100)  will  be  dropped;  that  is,  a 
balance  of  six  will  be  added.  The  new  total  is  there- 
fore 501  +  6,  or  507  (average  101).  In  the  same  way 
the  succeeding  totals  and  averages  may  be  derived.^ 

^  Mathematicians  sometimes  prefer  a  more  complex  form  of  the 
moving   average  known  as  the   progressive   mean.     This  is   similar   to 


TRENDS  AND  CYCLES  105 

The  two  moving  averages  just  described,  and  the 
index  on  which  they  are  based,  are  plotted  in  Figure 
7.  The  figure  will  serve  to  make  clear  the  general  rule 
that  the  more  inclusive  the  moving  average,  the 
smoother  the  trend  will  be.  A  disadvantage  of  the 
method  will  also  be  observed.  The  moving  average 
derived  from  a  given  series  of  items  will  always  be 
shorter  than  the  series;  and  the  more  inclusive  it  is, 
the  shorter  it  will  be.  It  is  possible,  however,  to  find 
a  tentative  substitute  for  the  lacking  items  in  the  trend 
by  repeating  the  extreme  items  in  finding  the  aver- 
ages. Thus,  in  the  foregoing  five-year  moving  average, 
a  trend  item  for  1911  might  have  been  obtained  as  fol- 
lows: 

2  X  100  +  95  +  108  +  100  ^  ^^^ 

5 

and  for  1910: 

2  X  100  +  2  X  95  -^  108 
5 

This  method  has  the  mathematical  advantage  of  mak- 
ing the  sum  of  the  trend  items  equal  to  the  sum  of 
the  data — a  fact  which  may  be  found  convenient  to 
use  as  a  basis  for  checking  the  computations.  It  may 
also  be  adapted  to  finding  a  current  trend  item  for  a 
series  of  index  numbers  that  is  kept  up  to  date. 

The  Line  of  Least  Squares.  If  a  straight-line  trend 
is  at  all  adapted  to  a  given  series,  the  most  satisfac- 
tory mathematical  trend  to  use  is  one  known  as  the 

the  moving  average,  as  LHustrat-ed  and  explained,  except  that  weights 
are  used  in  taking  the  average.  The  weights  are  derived  by  the  binomial 
theorem,  and  are  the  frequencies  of  a  theoretical  curve  of  distribu- 
tion. Thus  in  taking  a  five-year  progressive  mean,  each  group  of  five 
terms  is  averaged  by  applying  to  the  terms  in  succession  the  weights 
1:4:6:4:1.  Similarly,  a  seven  year  progressive  mean  would  make  use 
of  the  weights  1:6:15:20:15:6:1  (cf.  Slichter,  Elementary  Mathematical 
Analysis,  p.  194). 


100 


106  INTRODUCTION  TO  ECONOMIC  STATISTICS 

line  of  least  squares.  The  name  is  derived  from  the 
fact  that  the  line  is  so  dra\\^l  that  the  square  of  the 
deviations  of  the  data  from  it,  as  measured  on  the 
ordinates,  is  always  a  minimum.     The  line  of  least 


s. 

o 

/T     ^U 

4  . 

/o      • 

J  . 

/     • 

2  - 

/  • 

/  . 

/  / 
// 

^        AT' 

0 

r                             X 

-/    . 

// 

> 

-2   . 

/o 

-j  . 

/  / 

/ 

/ 

u/ 

T'/ 

^O 

-s  . 

y'         ... 

1      1 

1 

1 

1 

•^     1       1       1       1       1 

•S-4 

-J 

-£ 

-/ 

0    /    a    5   4-   s 

FiouEE  8.     Straight-line  trend.     TT',   line   of   least  squares  for  the 
seven  points  indicated;  UU',  line  of  unit  slope. 

squares  should  be  thoroughly  understood,  not  only 
because  of  its  usefulness  as  a  trend,  but  also  because 
it  is  the  basis  from  which  the  principal  method  of  com- 
puting correlation  is  developed.  In  explaining  it,  the 
following  simple  illustration  will  be  taken : 


TRENDS  AND  CYCLES  107 

Suppose  that  it  is  required  to  find  a  straight-line 
trend  for  the  index  y  (see  next  page),  the  average  of 
which  is  zero.  The  data  are  plotted  upon  coordinate 
paper — the  x  and  y  scales  having  preferably  the  same 
unit — as  shown  in  Figure  8.  The  average  of  the  data 
is  made  to  fall  on  the  x-axis,  and  the  middle  item  is 
plotted  on  the  y-axis.  If  we  consider  the  vertical 
distance  of  each  index  from  the  x-axis  to  represent  a 
force  bearing  upon  that  line,  then  the  total  moments 
of  these  forces  will  be  expressed  by  the  sum  of  the 
xy  's.  This  sum  may  be  compared  with  the  sum  of  the 
moments  of  a  line  passing  through  the  same  ordinates, 
and  forming  an  angle  of  forty-five  degrees  with  the 
X  and  y  axes  at  their  point  of  intersection.^  Such  a 
line  is  said  to  have  a  unit  slope;  that  is,  it  rises  one 
unit  {y)  for  each  unit  {x)  to  the  right.  Its  slope  may 
also  be  expressed  by  saying  that  the  tangent  of  its 
angle  (UOX)  is  unity.  The  sum  of  its  xy^s  is,  of 
course,  identical  with  the  sum  of  its  ic's  squared.  The 
slope  of  the  line  of  least  squares  is  found  by  compar- 
ing the  moments  of  the  data  with  those  of  the  line 
of  unit  slope,  as  expressed  in  the  formula : 

s  =  ^^^ 


in  which, 

S  =  slope  of  the  line  of  least  squares,  or  tangent  of 
its  angle 

X  ^  position  of  items  relative  to  middle  ordinate 

y  =  items,  as  given 

The  data  (y),  their  moments  (xy),  the  moments 
of  the  line  of  unit  slope  (x-),  and  the  computation  of 

*  The  angle  will  vary,  of  course,  if  the  x  and  y  scales  differ. 


108  INTRODUCTION  TO  ECONOMIC  STATISTICS 

the  line  of  least  squares,  are  shown  in  the  following 
table : 

y  X  X*  xy  Trend 


-3 

-3 

9 

9 

-4.5 

-4 

-2 

4 

8 

-3 

-2 

-1 

1 

2 

-1.5 

-1 

0 

0 

0 

0 

1 

1 

1 

1 

1.5 

5 

2 

4 

10 

3 

4 

3 

9 

12 

4.5 

=  0 

28 

S  = 

)     42 
=  1.5 

The  last  column  headed  ** trend"  gives  the  line  of  least 
squares.  This  column  is  computed  from  the  middle  or- 
dinate (0)  by  adding  the  slope  (S  =  1.5)  once  for  each 
successive  ordinate  in  a  positive  direction,  and  sub- 
tracting it  once  for  each  successive  ordinate  in  a  nega- 
tive direction. 

Data  having  an  average  of  zero  have  here  been 
taken  merely  for  the  sake  of  simplicity;  the  same 
process  w^th  but  little  modification  may  be  applied  to 
any  values.  The  dates  or  other  numbers  correspond- 
ing to  the  items  cannot  be  used,  however,  but  must 
be  replaced  by  an  x  scale  centering  at  the  middle  point 
of  the  series.  The  average  of  the  data  is  found, 
and  the  trend  is  computed  from  this  average  by  suc- 
cessive additions  of  the  slope  in  a  positive  direction, 
and  subtractions  in  a  negative  direction.  The  method 
is  illustrated  by  the  use  of  the  following  data,  which 
parallel  those  used  in  the  preceding  illustration,  except 
that  the  average  is  100.    This  increase  in  the  values  of 


TRENDS  AND  CYCLES  109 

y  disappears  from  2xy  because  it  affects  equally  both 
the  minus  and  the  plus  items. 


Year 

y 

X 

X2 

xy 

Trend 

1900 

97 

-3 

9 

-291 

95.5 

1901 

96 

-2 

4 

-192 

97 

1902 

98 

-1 

1 

-98 

98.5 

1903 

99 

0 

0 

0 

100 

1904 

101 

1 

1 

101 

101.5 

1905 

105 

2 

4 

210 

103 

1906 

104 

3 

9 

312 

104.5 

7 

)700 

28 

)  42 

(average) 


A  =  100  S  =  1.5 

The  following  details  may  be  noted:  (a)  If  there  are 
an  even  number  of  ordinates,  the  y-axis  will  lie  mid- 
way between  the  two  middle  ordinates,  which  are  num- 
bered as  -0.5  and  0.5  respectively.  The  horizontal 
positive  scale  will  therefore  read  0.5,  1.5,  2.5,  etc.,  and 
the  negative  scale  will  be  the  reverse,  (b)  It  will  some- 
times be  found  that  the  value  of  ^  is  negative.  This 
indicates  a  downward  slope  of  the  line  of  least  squares, 
(c)  The  position  of  the  line  of  least  squares  is  de- 
scribed by  designating  the  period  coinciding  with  the 
y  axis  as  the  point  of  origin,  and  by  expressing  the 
value  of  y  algebraically  in  terms  of  the  average  and 
the  slope.  Thus  in  the  above  illustration  the  point  of 
origin  is  1903,  and  the  equation  of  the  trend  is  y  =  100 
-f  1.5x.i 

^  It  has  been  suggested  that  the  method  of  least  squares  might  be 
applied  to  the  finding  of  a  price  index  {Quarterly  Journal  of  Economics, 
August,  1921,  page  567).  The  expenditure  for  any  given  commodity 
may  be  plotted  on  a  coordinate  chart  as  the  value  of  y,  and  the  num- 
ber of  units  bought  may  be  plotted  as  the  value  of  x.  The  slant  of  a 
line  (tangent  of  the  base  angle)  drawn  from  the  intersection  of  the 
axes  to  the  point  determined  by  the  values  of  x  and  y,  represents 
the   price.     The   average   price   of   a   number   of   commodities   may   be 


110  INTRODUCTION  TO  ECONOMIC  STATISTICS 

The  Parabola  Trend.  A  broad  treatment  of  the  sub- 
ject of  curve  fitting  would  lead  the  student  beyond  the 
range  of  ordinary  statistical  work.  We  shall  not, 
therefore,  follow  the  subject  farther,  except  to  take  up 


/2o/  \ 


m. 


m^ 


90. 


io. 


70. 


60^ 
/870 


1 

mo 


mo 


mo 


/m 


Figure  9.     Index  of  wholesale  prices  in  the  United  States  (see  Table 
X)   and  trend.     Indexes  for  1870-1880  converted  to  a  gold  basis. 

a  simple  method  of  adjusting  a  parabola  to  an  index. 
The  method  is  one  which  is  often  used  by  engineers, 
and  has  also  recently  come  into  use  to  some  extent 
among  statisticians. 

taken  as  the  slant  of  the  line  of  least  squares  determined  by  all  the 
coordinate  pairs  of  x  and  y,  and  having  the  intersection  of  the  axea 
as  the  point  of  origin  (y  =  Sx).  In  such  a  case,  of  course,  all  the 
values  of  xy  will  be  positive.  To  give  definite  comparisons  at  different 
dates,  this  method  would  require  the  use  of  "dollar's  worths"  as 
physical  units.  While  the  method  is  ingenious,  it  is  of  questionable 
validity,  since  in  effect  it  involves  a  weighting  of  the  prices  by  the 
square  of  the  quantities  in  the  process  of  finding  the  average.  A 
defense  of  tlic  nietliod  on  the  basis  of  the  use  of  least  squares  in  the 
theory  of  errors  docs  not  appear  to  be  valid,  since  the  theory  of  errors 
would  call  for  merely  the  arithmetic  mean  of  the  number  of  determina- 
tiona. 


TRENDS  AND  CYCLES  111 

The  method  of  adjusting  a  parabola  will  be  illus- 
trated by  applying  it  to  the  Bureau  of  Labor  Statistics ' 
wholesale  price  index  for  the  years  1896  to  1915  inclu- 
sive, as  charted  in  Figure  9.  This  figure  includes  also 
the  same  index  for  the  years  1870-1895  (gold  prices), 
to  which  a  line  of  least  squares  has  been  fitted.  But  it 
may  be  easily  seen  that  a  similar  trend  would  not  be 
suited  to  the  succeeding  index  numbers.  The  some- 
what regular  curve  of  the  latter  portion  of  the  index 
indicates  that  a  parabola  of  the  second  order  would 
be  appropriate. 

In  fitting  the  parabola,  the  year  1895  has  been  taken 
as  the  point  of  origin,  though  this  year  is  not  included 
in  the  results.  It  is  estimated  by  inspection  of  the 
graph  that  the  trend,  if  extended  to  1895,  would  have 
a  value  of  64  at  that  date.  Two  other  points  deter- 
mining the  trend  may  be  similarly  located,  one  at 
about  the  middle  of  the  series  and  one  at  the  end. 
These  points  have  been  taken  as  88  for  the  year  1905 
(x  =  10),  and  102  for  the  year  1915  (x  =  20).  We 
have,  then,  these  coordinate  values  of  x  and  y: 

If  X  r-    0,        y  =     64 

If  X  =  10,        y  =    88 
If  X  =  20,        y  =  102 

The  equation  of  a  parabola  of  the  second  order  is, 
y  =  a  -f-  bx  +  cx^ 

If  the  coordinate  values  of  x  and  y  as  just  stated  are 
substituted  successively  in  this  equation,  the  following 
results  will  be  obtained : 

64  =  a 

88  =  a  +  10b  +  100c 
102  =  a  +  20b  +  400c 


112  INTRODUCTION  TO  ECONOMIC  STATISTICS 

Solving  for  the  constants  gives, 

a  =  64 

b=    2.9 

c=   -.05 
Substituting  these  values  in  the  original  equation  gives 
the  equation  of  the  required  trend, 

y  =  64  +  2.9x  -.OSx^ 
from  which  the  value  of  each  item  in  the  trend  may  be 
found  by  substituting  the  coordinate  value  of  x. 

Since  trends  are  used  as  a  basis  for  measuring  fluc- 
tuations, the  deviations  of  the  data  from  the  trend 
are  usually  computed.  This  is  done  by  subtracting 
each  item  in  the  trend  from  the  corresponding  item 
in  the  data.  If  the  trend  has  been  accurately  con- 
structed, the  positive  and  negative  deviations  should 
be  practically  equal ;  that  is,  their  sum  should  be  zero. 
Where  the  parabola  has  been  used,  a  certain  error  will 
probably  be  found  to  have  resulted  from  the  fact  that 
the  original  points  determining  the  curve  were  located 
merely  by  inspection.  An  adjustment  (centering)  to 
remove  this  error  may  be  made  by  finding  the  sum  of 
the  deviations  (2D),  dividing  it  by  the  number  of  the 

items  (N),  adding  the  result f-^j  to  each  of  the  trend 

items,  and  subtracting  it  from  each  of  the  deviations. 
This  is  expressed  in  the  equation  of  the  trend  simply 

by  adding  the  correction  (  -^  j  to  the  value  of  a.    In 

work  in  correlation,  however,  the  correction  may  be 
more  easily  made  by  another  method,  as  will  be  evi- 
dent later.  When  comparisons  of  the  fluctuations  in 
different  scries  are  to  be  made,  either  by  graphing  or 


TRENDS  AND  CYCLES 


113 


by  the  computation  of  a  coefficient  of  correlation,  it  is 
often  necessary  to  find  the  standard  deviation.  For 
graphic  representation,  the  deviations  are  reduced  to 
multiples  of  the  standard  deviation,  which  serves  as  a 
comparable  unit.    Table  XVI  shows  the  derivation  of 

TABLE  XVI 


DEEIVATION  OF  TREND  AND  CYCLES  OF  WHOLESALE  PRICES, 

BASED  ON  BUREAU  OF  LABOR  STATISTICS  INDEX, 

UNITED  STATES,  1896-1915 


Equation  of  trend,    y  =  64  +  2.9x  —  .05x^ 
Equation,  corrected,  y  =  63.825  +  2.9x  —  ,05x» 


Point  of  origin,  1895. 


TEAR 

PRICE 

TREND 

D 

D 

D» 

CYCLES • 

INDEi 

X 

,y 

CENTERED 

D/a 

1896 

66 

1 

66.85 

-.85 

-.68 

.4624 

-.33 

1897 

67 

2 

69.60 

-2.60 

-2.42 

5.8564 

-1.17 

1898 

69 

3 

72.25 

-3.25 

-3.08 

9.4864 

-1.49 

1899 

74 

4 

74.80 

-.80 

-.62 

.3844 

-.30 

1900 

80 

5 

77.25 

2.75 

2.92 

8.5264 

1.41 

1901 

79 

6 

79.60 

-.60 

-.42 

.1764 

-.20 

1902 

85 

7 

81.85 

3.15 

3.32 

11.0224 

1.60 

1903 

85 

8 

84.00 

1.00 

1.18 

1.3924 

.57 

1904 

86 

9 

86.05 

-.05 

.12 

.0144 

.06 

1905 

85 

10 

88.00 

-3.00 

-2.82 

7.9524 

-1.36 

1906 

88 

11 

89.85 

-1.85 

-1.68 

2.8224 

-.81 

1907 

94 

12 

91.60 

2.40 

2.58 

6.6564 

1.25 

1908 

91 

13 

93.25 

-2.25 

-2.08 

4.3264 

-1.00 

1909 

97 

14 

94.80 

2.20 

2.38 

5.6644 

1.15 

1910 

99 

15 

96.25 

2.75 

2.92 

8.5264 

1.41 

1911 

95 

16 

97.60 

-2.60 

-2.42 

5.8564 

-1.17 

1912 

101 

17 

98.85 

2.15 

2.32 

5.3824 

1.12 

1913 

100 

18 

100.00 

0 

.18 

.0324 

.09 

1914 

100 

19 

101.05 

-1.05 

-.88 

.7744 

-.43 

1915 

101 

20 

102.00 

-1.00 

-.82 

.6724 

-.40 

16.40         17.92  20)85.9880 
-19.90       -17.92      


20)-3.50 
K  =  -  .175 


0.     a'  =  4.2994 
(J  =  2.07 


8.66 
-8.66 


^  Any  set  of  deviations  taken  from  an  average  or  trend,  and  intended 
to  measure  cyclic  movements,  are  commonly  designated  as  cycles.  They 
need  not  necessarily  be  reduced  to  units  of  the  standard  deviation.  In 
some  cases  no  complete  cyclic  movement  may  be  discovered,  but  the 
same  designation  may  be  used. 


114  INTRODUCTION  TO  ECONOMIC  STATISTICS 

the  trend  just  discussed,  together  with  the  correction, 
and  the  reduction  of  the  deviations  to  multiples  of  the 
standard  deviation.  Figures  10  and  11  show  price 
cj^cles  as  thus  measured  compared  mth  other  cycles 
similarly  obtained. 

The  parabola  may  be  used  where  compound  curves 
are  required  by  adding  the  term  dx^,  and  perhaps  ex* 
to  the  equation.    For  each  term  thus  added,  an  addi- 


1900 


I     I     I 

1905 


I    I    I    i    I 

I9IS 


Figure  10.  Cycles  of  wholesale  prices  (solid  line)  and  commitments 
to  New  York  State  prisons  (broken  line). 

Eeprinted,  with  permission,  from  Tlie  Quarterly  Journal  of  the  Uni- 
versity of  North  Dakota,  January,   1922. 


tional  point  may  be  located  by  inspection,  and  the 
trend  may  thus  be  more  exactly  fitted.  But  the  work 
of  solving  and  applying  such  equations  becomes  very 
laborious.  With  practice,  however,  the  student  will 
find  ways  of  abbreviating  the  process  and  modifying 
it  to  suit  his  purposes.  Often  the  terms  of  the  equa- 
tion may  be  estimated  by  simple  experimentation.  The 
position  of  the  point  of  origin  may  be  varied  to  suit 
given  requirements.    A  compound  curve  shaped  some- 


TRENDS  AND  CYCLES 


115 


what  like  an  italic  /  may  be  obtained  by  using  only  odd 
numbered  powers  of  x  in  the  equation,  and  taking  the 
point  of  origin  near  the  middle  of  the  original  series. 
"With  a  little  ingenuity,  sine  curves  and  other  trends 
may  be  experimentally  fitted.^ 


Figure  11.  Cycles  of  wholesale  prices  (solid  line)  and  marriage 
rate   (broken  line)   in  the  United  States. 

Eeprinted,  with  permission,  from  The  Quarterly  Journal  of  the  Uni- 
versity of  North  Dakota,  January,  1922. 

Analyzing  Business  Barometers.  A  somewhat  in- 
tricate problem  in  trends  is  met  when  monthly  data 
are  employed  as  indexes,  or  barometers,  of  business 
conditions.  Since  such  barometers  are  very  generally 
consulted  as  guides  to  business  activities,  their  inter- 

*  When  it  appears  that  the  trend  of  an  index  series  increases  or 
decreases  by  approximately  a  fixed  ratio,  an  exponential  curve  may 
be  readily  fitted  as  follows.  Find  the  logarithms  of  the  data,  plot 
them,  and  construct  a  straight-line  trend  by  means  of  the  semi-aver- 
ages, or  by  a  line  of  least  squares.  Eead  the  items  of  the  trend  from 
the  chart,  consider  them  to  be  logarithms,  and  find  the  corresponding 
numbers.  These  numbers  will  be  the  items  of  the  required  trend.  In 
a  long  series  the  work  may  be  abbreviated  by  grouping  the  original 
items,  aa  by  decades,  and  fitting  the  curve  to  the  averages  of  these 
groups. 


116  INTRODUCTION  TO  ECONOMIC  STATISTICS 

pretation  is  a  matter  of  great  practical  importance. 
The  difficulty  involved  in  their  use  lies  in  the  com- 
plexity of  the  influences  playing  upon  them.  For  con- 
venience of  analysis  these  influences  have  been  clas- 
sified as  (1)  a  seasonal  variation  usually  due  to  the 
dependence  of  industry  upon  weather  conditions,  illus- 
trated by  the  rising  of  the  interest  rate  mth  the  move- 
ment of  crops,  (2)  a  cyclic  movement  covering  an  in- 
terval of  several  years  and  marked  by  alternating  in- 
dustrial depression  and  activity,  and  (3)  a  secular 
trend  or  gradual  change  due  in  most  cases  to  the 
growth  movement,  as  seen  in  the  increase  in  produc- 
tion. In  addition  to  influences  which  may  be  appro- 
priately classed  under  one  of  these  three  headings, 
there  are  others  that  must  be  looked  upon  as  more  or 
less  accidental  interruptions,  of  which  no  exact  account 
can  be  taken. 

Measuring  Seasonal  Variations.  The  most  sat- 
isfactory method  of  analyzing  monthly  business  ba- 
rometers is  first  to  compute  an  index  of  seasonal  va- 
riations, and  then  to  subtract  it,  month  by  month, 
from  an  index  of  the  data  based  upon  the  secular 
trend.  The  result  is  an  index  of  the  cycles.  As  has 
already  been  noted,  the  twelve-month  moving  average 
is  sometimes  assumed  to  measure  the  data  as  distinct 
from  the  seasonal  variations.  But  such  a  method  of 
elimination  takes  into  account  the  fluctuations  of  only 
one  year  at  a  time.  Other  methods  have  therefore 
been  resorted  to  with  the  purpose  of  measuring  the 
seasonal  swing  more  exactly  on  the  basis  of  several 
years.  A  simple  method  of  this  sort,  which  may  be 
considered  valid  for  a  period  in  which  the  cyclic  in- 


TRENDS  AND  CYCLES 


117 


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118  INTRODUCTION  TO  ECONOMIC  STATISTICS 

fluences  are  moderate  and  well  distributed  along  an 
approximately  straight-line  trend,  is  illustrated  in 
Table  XVII. 

The  Method  of  Averages.  The  problem  stated  and 
solved  in  the  table  is  the  finding  of  the  seasonal  varia- 
tions in  the  interest  rate  on  the  somewhat  slender  basis 


la^Q 


I8  60 


18  80 


I900 


1920 


Figure  12.  Average  interest  rate  on  commercial  paper  in  the  United 
States  (lower  line)  compared  with  index  of  wholesale  prices  (upper 
line). 

Adapted  from  Monthly  Review,  New  York  Federal  Reserve  Bank. 


of  the  five  years  1909  to  1913,  inclusive.  The  monthly 
rates  as  given  are  first  averaged  both  by  columns  and 
rows  in  order  to  obtain  the  annual  averages  and  the 
month  averages.  The  process  may  therefore  be  called 
the  method  of  averages. 

If  the  interest  rate  had  maintained  approximately 
the  same  level  from  year  to  year  during  the  interval 


TRENDS  AND  CYCLES  119 

studied,  nothing  more  would  be  needed  than  to  reduce 
the  month  averages,  or  types,  to  a  percentage  of  their 
own  annual  average.  The  results  would  form  an  index 
of  the  seasonal  variations.  But  if  there  is  a  secular 
movement,  it  must  be  canceled  from  such  an  index. 
If  an  interval  of  half  or  three-quarters  of  a  century 
is  taken  into  account,  a  general  downward  trend  of  the 
interest  rate  may  be  discovered,  as  may  be  seen  in 
Figure  12.  But  the  five  years  here  studied  happen  to 
be  an  exception.  By  means  of  the  line  of  least  squares, 
a  positive  slope  of  0.019  monthly  is  revealed.  By  ap- 
plying this  slope  to  the  average  of  the  month  types, 
an  annual  trend  is  constructed,  having  its  point  of 
origin  midway  between  the  June  and  July  items.  It 
will  be  noted  that  one-half  the  slope  must  be  added  to 
the  average  to  obtain  the  July  trend  item,  and  the  same 
amount  subtracted  to  obtain  the  June  item.  In  ob- 
taining the  remaining  trend  items,  the  slope  is  applied 
as  previously  explained. 

Confusion  may  perhaps  here  arise  from  the  fact 
that  the  slope  was  computed  from  the  five  annual  aver- 
ages, but  was  applied  to  the  construction  of  a  trend 
with  which  to  compare  the  monthly  data.  But  it  should 
be  obvious  that  in  this  case  a  slope  obtained  from  the 
month  averages  would  be  materially  affected  by  the 
seasonal  swing.  This  we  wish  to  retain,  while  the  secu- 
lar trend  we  wish  to  cancel.  Of  course  we  could  find 
the  secular  trend  from  the  monthly  data  taken  consecu- 
tively as  sixty  items,  but  such  a  method  would  be  un- 
necessarily laborious.  Hence  we  find  it  from  the  five 
annual  averages,  and  apply  it  to  the  construction  of 
a  line  that  will  serve  to  cancel  the  secular  trend  from 


120  INTRODUCTION  TO  ECONOMIC  STATISTICS 

the  month  types.  This  cancellation  is  accomplished  by 
dividing  the  month  averages,  item  by  item,  by  the 
trend.  The  quotients  should  be  centered,  if  necessary, 
by  reducing  them  to  percentages  of  their  common  aver- 
age. The  result  is  an  index  of  seasonal  variations, 
from  which  the  percentage  deviations,  month  by  month, 
may  be  directly  stated.^ 

Applying  the  Seasonal  Index.  The  method  of 
applying  the  seasonal  index  has  already  been  sug- 
gested, and  may  be  described  as  follows.  A  secular 
trend  for  the  whole  period  under  consideration  is  con- 
structed— in  this  case  the  line  of  least  squares  already 
found  may  be  extended — and  the  data  are  reduced 
month  by  month  to  percentages  of  the  trend.  From 
each  month's  item  as  thus  found  the  seasonal  index 
for  the  same  month  is  subtracted.  The  remainders 
are  assumed  to  measure  the  cycles,  and  may  be  plotted 
as  deviations  above  and  below  a  horizontal  axis.  The 
seasonal  index  may,  with  caution,  be  applied  to  other 
years  than  those  from  which  it  is  derived;  of  course, 
the  greater  the  number  of  normal  years  from  which 
it  is  derived,  the  safer  such  an  extension  of  its  use 
to  comparable  years  becomes. 

The  Link-relative  Method.  A  complex  but  more  ac- 
curate method  for  measuring  seasonal  variations  has 

^  Another  variation  of  the  method  of  averages — one  that  is  per- 
haps theoretically  preferable,  but  which  involves  more  extended  cal- 
culations— may  be  briefly  described  as  follows:  A  twelve-month 
moving  average  of  the  monthly  data  tlirough  a  given  scries  of  years 
is  first  ^ound.  This  is  adjusted  to  make  it  conform  to  the  ordinatea 
of  the  original  series  by  deriving  from  it  a  two-month  moving  average. 
There  is  then  obtained,  for  each  month^  the  ratio  of  the  original  monthly 
item  to  the  corresponding  adjusted  moving  average.  The  median  of 
the  ratios  so  obtained  for  the  Januaries  is  taken  as  the  index  of  sea- 
sonal variation  for  January;  and  index  numbers  for  the  other  months 
are  similarly  obtained.     The  twelve  results  are  then  centered,  if  neces- 


TRENDS  AND  CYCLES  121 

been  developed  by  Professor  Persons,  and  applied  to 
the  analyses  appearing  in  the  early  numbers  of  the 
Review  of  Economic  Statistics.    In  a  somewhat  sim- 
plified   form,    this    method    is    illustrated    in    Table 
XVII- A,  which  is  based  on  the  data  of  Table  XVII. 
Briefly  stated,  the  method  consists  in  finding  what  are 
called  '' link-relatives";  that  is,  the  percentage  which 
the  index  of  each  month  is  of  the  preceding  month. 
The  median  link-relatives  are  then  selected  from  each 
month's  series,  and  are  tabulated  as  the  month  types.^ 
Beginning  with  December  as  a  base  (100%),  the  types 
are  multiplied  consecutively,  producing  an  index  series 
from  January  to  December  for  a  typical  year.    If  the 
final  December  item  fails  to  come  out  to  100%,  in  con- 
formity with  the  base  in  the  preceding  December,  a 
secular  trend  is  evidently  disturbing  the  index.    The 
discrepancy,  if  moderate,  may  be  removed  by  distribut- 
ing it  throughout  the  year ;  that  is,  by  subtracting  one- 
twelfth  of  it  from    the    January  index,  two-twelfths 
from  the  February  index,  and  so  on  through  the  year.^ 
The  results  are  designated  an  adjusted  index.    The  ad- 
justed items  are  next  centered  by  reducing  them  to 
a  base  of  the  average  of  the  series.     In  making  the 
computations  the  figures  were  carried  to  one  more 
place  than  is  shown  in  the  table. 

The  superiority  of  this  method  lies  in  the  fact  that 
in  taking  the  link-relatives,  and  in  selecting  their 
medians  as  the  month  types,  the  effects  of  the  cyclic 

sary,  by  reducing  them  to  percentages  of  their  common  average  (Cf. 
Jordan,  Biisiness  Forecasting,  p.  212). 

*  To  get  the  best  results,  the  median  should  be  based  on  a  larger 
number  of  years  than  are  here  taken. 

'A  more  exact  method  is  to  apportion  the  discrepancy  geometrically; 
that  is,  to  divide  the  January  index  by  the  twelfth  root  of  the  final 
December  index  (written  as  a  decimal),  the  February  index  by  the 
square  of  this  root,  and  so  on. 


122  INTRODUCTION  TO  ECONOMIC  STATISTICS 


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TRENDS  AND  CYCLES  123 

movements  and  of  chance  influences  are  minimized. 
The  secular  trend  is  satisfactorily  eliminated  by  the 
process  of  adjusting.  The  index  thus  obtained  should 
be  accurate  for  the  years  just  preceding  the  establish- 
ment of  the  Federal  Reserve  System.  Since  that  time 
seasonal  changes  have  lessened.^ 

Business  Cycles.  The  Review  of  Economic  Statis- 
tics, in  the  analyses  just  referred  to,  has  made  exhaus- 
tive studies  of  the  cyclic  movements  of  the  commonly 
used  business  barometers.  After  measuring  the  in- 
fluence of  seasonal  variations  by  a  process  similar  to 
that  just  described,  it  determined  the  cycles  on  the 
basis  of  lines  of  least  squares.  It  then  combined 
twelve  of  the  principal  barometric  indexes  into  three 
composite  indexes  which  are  taken  as  measurements  of 
speculative  activity,  business  activity,  and  banking 
strain,  respectively.  A  chart  of  these  three  composite 
indexes  for  the  years  1903  to  1913,  inclusive,  is  here 
reproduced  (Figure  13).  The  chart  shows  very  clearly 
the  general  stages  of  the  business  cycle,  from  the  pre- 
dominance of  the  speculative  activity  which  marks  the 
awakening  from  a  period  of  depression,  through  the 
period  of  intensified  production,  and  into  the  period 
of  banking  strain  which  heralds  another  depression. 
The  various  barometric  series  used  in  constructing  the 
figure  are  indicated  in  the  explanation  accompanying 
the  title.  It  should  be  added  that  while  New  York  bank 
loans  point  to  the  speculative  aspects  of  the  cycle,  those 
outside  New  York  conform  more  closely  to  business 

*For  the  most  exhaustive  study  of  seasonal  variations  in  the  interest 
rate,  the  student  should  consult  "Seasonal  Variations  in  the  Relative 
Demand  for  Money  and  Capital  in  the  Fnited  States"  (National 
Monetary  Commission,  1910),  by  E.  W.  Kemmerer,  See  also  the 
Monthly  Review,  Federal  Reserve  Bank  of  New  York,  Feb.  1,  1922. 


124  INTRODUCTION  TO  ECONOMIC  STATISTICS 

activity.  For  practical  purposes  measurements  of  the 
general  aspects  of  the  cycle  need  to  be  supplemented 
by  indexes  showing  the  position  of  particular  indus- 
tries, as  indicated  by  relative  production  and  stocks  of 
goods  on  hand.  This  need  is  now  being  met  in  part  by 
the  Federal  Reserve  Bank  of  New  York,  and  by  other 
agencies. 

Several  phenomena  more  psychological  than   eco- 
nomic in  nature  show  a  tendency  to  fluctuate  more  or 


Figure  13.     The  index  of  general  business  conditions,  1903-14. 

A,  Speculation:  New  York  Bank  clearings,  average  price  of  indus- 
trial stocks,  average  price  of  railroad  stocks,  and  average  price  of 
railroad  bonds. 

B,  Business:  Bank  clearings  of  the  United  States  outside  New  York 
City,  Bradstrect's  index  of  wholesale  commodity  prices,  United  States 
Bureau  of  Labor  Statistics'  index  of  wholesale  commodity  prices,  and 
pig-iron  production. 

C,  Banking:  Interest  rates  on  60-90  day  and  on  4-6  months  com- 
mercial paper  in  New  York  City,  loans  and  deposits  of  New  York  City 
clearing  house  banks  (both  inverted). 

Reproduced  from  Beview  of  Economic  Statistics,  by  permission  of 
the  editors. 


less  closely  with  the  business  cycle.  Professor  A.  H. 
Hansen  has  recently  showTi  statistically  that  since 
1898  strikes  increased  with  prosperity,  though  before 
that  date,  when  the  trend  of  prices  was  downward, 
they  increased  with  depression  (cf.  American  Eco- 
nomic Review,  Dec,  1921,  pp.  617-621).  Mr.  Roger 
Babson  has  traced  a  connection  between  church  growth 
and  the  business  cycle;  religious  activity  being  ap- 
parently intensified  during  a  period  of  depression. 


TRENDS  AND  CYCLES  125 

Unemployment,  failures,  suicides,  and  crime  generally, 
also  increase  during  a  period  of  depression.  On  the 
other  hand,  immigration,  the  marriage  rate,  and  ex- 
travagance, tend  to  increase  with  a  period  of  prosper- 
ity (cf.  Figures  10  and  11,  pages  114  and  115). 

As  an  example  of  the  application  of  statistical  meth- 
ods to  the  practical  problem  of  forecasting  the  cyclic 
movements  of  business,  the  ''Annalist  Barometer  and 
Business  Index  Line"  may  be  cited.  This  index  is 
published  in  graphic  form,  together  with  a  brief  ex- 
planation, each  week  in  the  Annalist,  a  well-known 
financial  paper  of  New  York.  It  is  derived  from  the 
data  elaborated  by  the  Review  of  Economic  Statis- 
tics, already  briefly  described.  The  index  is  the  recip- 
rocal of  a  weighted  average  of  the  deviations  from 
normal  of  commodity  prices,  interest  rates,  pig  iron 
production.  New  York  bank  clearings,  and  bank  clear- 
ings outside  of  New  York.  Since  these  series  of  data, 
when  directly  combined,  measure  the  later  phases  of 
the  business  cycle,  their  decline  precedes  and  fore- 
casts the  rise  in  stocks  marking  the  beginning  of  the 
next  cycle;  and  their  rise  similarly  forecasts  a  decline 
in  stocks.  By  taking  the  reciprocal  of  the  deviations 
of  the  combined  series,  the  forecast  is  made  direct 
instead  of  inverse.  By  a  comparison  of  the  forecast- 
ing index  and  the  movement  of  stocks  in  former  years, 
the  decisiveness  of  change  in  the  former  necessary  to 
constitute  a  forecast  of  the  latter  has  been  determined. 
A  detailed  account  of  the  construction  and  use  of  the 
index  will  be  found  in  the  Annalist  of  March  28  and 
of  October  24,  1921. 

The  student  who  desires  to  inquire  more  intensively 


126  INTRODUCTION  TO  ECONOMIC  STATISTICS 

into  the  statistics  of  the  business  cycle  should  consult 
for  himself  the  data  published  in  the  Review  of  Eco- 
nomic Statistics  already  mentioned.  In  addition  he 
should  become  famihar  wdth  Wesley  C.  Mitchell's 
standard  work  on  Business  Cycles,  and  with  a 
more  recent  work  by  D.  F.  Jordan  on  Business  Fore- 
casting. In  connection  with  the  underlying  causes 
of  the  cycle,  reference  should  be  made  to  the  interest- 
ing but  very  technical  works  of  H.  L.  Moore  (cf.  Eco- 
nomic Cycles,  and  articles  in  the  Quarterly  Journal 
of  Economics,  February,  August,  and  November, 
1921).  Professor  Moore  discovers  some  relation  to 
exist  between  a  weather  cycle  of  heavier  and  lighter 
rainfall  and  the  business  cycle,  the  average  duration  of 
each  cycle  being  about  eight  years.  The  moist  years 
bring  as  a  rule  larger  crops,  with  some  tendency  to  a 
lowering  of  the  general  price  level,  followed  during 
the  drier  years  by  a  rise  in  the  price  level.  This  rela- 
tion seems  to  be  clearer  for  English  prices  than  for 
American.  The  weather  cycle  is  shown  to  be  synchro- 
nous in  several  countries,  and  to  be  correlated  with  a 
cycle  of  barometric  pressure,  which  in  turn  may  have 
astronomical  causes.  But  while  the  subject  is  very 
interesting  and  valuable  theoretically,  the  correlations 
disclosed  are  too  irregular  to  be  of  great  practical 
value. 

REFERENCES 

Babson,  Roper  W.,  Business  Barometers. 

Davies,  G.  R.,  "Social  Aspects  of  the  Business  Cycle,"  Qimr- 

ierly  Journal  of  the  University  of  North  Dakota,  January, 

1922. 
Hurlin,  Ralph  G.,  "The  Long-Time  Trend  of  Prices  in  the 

United  States,"  The  Annalist,  July  4,  1921. 


TRENDS  AND  CYCLES  127 

Jordan,  D.  F.,  Business  Forecasting. 

Kemmerer,  E.  W.,  High  Prices  and  Deflation. 

Mitchell,  Wesley  C.,  Business  Cycles. 

Moore,  Henry  L.,  Economic  Cycles:  Their  Law  and  Cause. 

Peddle,  John  B.,  The  Construction  of  Graphical  Charts, 
Chapter  VI. 

Persons,  W.  W.,  "Construction  of  a  Business  Barometer 
Based  upon  Annual  Data,"  American  Economic  Review, 
December,  1916,  pp.  739-769. 

Piatt,  Andrew  A.,  National  Monetary  Commission. 

Tingley,  Richard  H.,  "Another  Yardstick  of  Banking  Condi- 
tions," The  Annalist,  November  28,  1921,  p.  511. 


EXERCISES 

1.  Plot  the  data  for  production  and  price  of  wheat,  1870- 
1920  (Tables  XIV  and  XIV-A,  pp.  76  and  77),  and 
draw  a  free-hand  trend  for  each  series. 

2.  As  in  Exercise  1,  construct  free-hand  trends  for  the  pro- 
duction and  price  of  com. 

3.  Plot  the  index  of  physical  production  of  crops,  1870- 
1920  (Table  XV,  p.  81),  and  draw  a  straight-line  trend 
by  inspection. 

4.  Apply  the  method  of  semi-averages  to  the  construction 
of  a  straight-line  trend  for  the  data  of  the  preceding 
exercise. 

5.  Compute  a  five-year  moving  average  of  per  capita  pro- 
duction, 1890-1918.  Plot  both  the  trend  and  the  data 
from  which  it  is  derived  on  17"x22"  cross-section  paper, 
and  measure  graphically  the  deviations  from  the  trend. 

6.  Plot  on  a  horizontal  axis  the  deviations  obtained  in 
the  preceding  exercise.  Find  the  average  deviation,  and 
indicate  this  on  the  graph  for  both  the  plus  and  the  minus 
deviations. 

7.  Compute  and  plot  a  straight-line  trend  (line  of  least 
squares)  for  the  following  price  index. 

Year  Prices  (3  articles) 

1897  85 

1898  70 

1899  90 

1900  130 

1901  125 


128  INTRODUCTION  TO  ECONOMIC  STATISTICS 

8.  Find  straight-line  trends  (lines  of  least  squares)  for 
the  production  and  price  of  pig  iron  as  given  below, 
taking  each  year  separately. 


PEODUCTION  AND  PRICE  OF  PIG  IRON     (Iron  Age) 
(000  omitted  from  production) 


Month 

1909 

1910 

1911 

1912 

1913 

Tons 

$ 

Tons 

$ 

Tons 

$ 

Tons 

$ 

Tons        $ 

Jan.  . 

1,797 

16.25 

2,608 

17.25 

1,759 

14.25 

2,057 

13.25 

2,795  16.95 

Feb.. 

1,707 

16.13 

2,397 

17.06 

1,794 

14.25 

2,100 

13.31 

2,586  16.69 

March 

1,832 

15.05 

2,617 

16.30 

2,188 

14.25 

2,405 

13.50 

2,763  16.31 

April . 

1,738 

14.25 

2,483 

15.37 

2,065 

14.25 

2,375 

13.75 

2,752  15.65 

May.. 

1,883 

14.50 

2,390 

15.00 

1,893 

13.95 

2,512 

14.15 

2,822  14.94 

June. 

1,930 

14.70 

2,265 

14.85 

1,787 

13.44 

2,440 

14.25 

2,628  14.06 

July.. 

2,103 

15.75 

2,148 

14.75 

1,793 

13.25 

2,410 

14.70 

2,560  13.75 

Aug.. 

2,248 

16.38 

2,106 

14.31 

1,926 

13.45 

2,512 

15.06 

2,543  14.06 

Sept- . 

2,385 

17.35 

2,056 

14.25 

1,977 

13.31 

2,463 

15.87 

2,505  14.25 

Oct.. 

2,599 

17.88 

2,093 

14.25 

2,102 

13.25 

2,689 

16.80 

2,546  14.35 

Nov.. 

2,547 

17.75 

1,909 

14.25 

1,999 

13.20 

2,630 

17.25 

2,233  13.87 

Dec. . 

2,635 

17.45 

1,777 

14.25 

2,043 

13.19 

2,782 

17.25 

1,983  13.95 

Total 

25,410 

16.12 

26,855 

15.16 

23,329 

13.67 

29,383 

14.93 

30,722  14.90 

(Reprinted,   with  permission,  from  Babson's  Desk  Sheet.) 


9.  Find  the  average  index  of  per  capita  physical  pro- 
duction in  the  United  States  (page  81)  for  each  dec- 
ade from  1870  to  1919.  Using  the  resulting  five  aver- 
ages, construct  a  line  of  least  squares.  Plot  the  original 
data  and  the  trend  thus  found. 

10.  Find  the  average  index  of  production  of  iron  and  copper 
by  five  year  periods  from  1870  to  1914.  Plot  these  aver- 
ages, and  fit  to  them  a  parabola  of  the  second  degree. 

11.  Compute  and  plot  the  cycles  of  the  interest  rate,  1909- 
1913,  using  the  data  and  index  of  seasonal  variations 
presented  in  Table  XVII,  page  117. 

12.  Using  the  data  given  below,  find  an  index  of  seasonal 
variations  in  exports  of  merchandise  (a)  by  the  method 
of  averages,  and  (b)  by  the  link-relative  method. 


TRENDS  AND  CYCLES  129 

EXPORTS  OF  MERCHANDISE,  UNITED  STATES,  1909-1913 
(In  Millions  of  Dollars) 

1909       1910      1911       1912      1913 

January    157  144  197  202  227 

February   126  125  176  199  194 

March  139  144  162  205  187 

April   125  133  158  179  200 

May 123  131  153  175  195 

June   117  128  142  138  163 

July  109  115  128  149  161 

August   110  135  144  168  188 

September   154  169  196  200  218 

October   201  208  210  255  272 

November   194  207  202  278  246 

December    172  229  225  250  233 

(December,  1908,  170) 

13.  The  following  table,  adapted  from  the  Yearbook  of  the 
Department  of  Agriculture,  1918,  gives  the  farm  price 
of  wheat  in  the  United  States  (cents  per  bushel)  on  the 
first  day  of  each  month  for  the  years  1909  to  1913 
inclusive.  Using  the  method  of  link  relatives,  derive  an 
index  of  seasonal  variations. 

1909         1910       1911       1912       1913 

January  1   . . . .  93.5  103.4  88.6  88.0  76.2 

February  1    ...  95.2  105.0  89.8  90.4  79.9 

March  1   103.9  105.1  85.4  90.7  80.6 

April  1 107.0  104.5  83.8  92.5  79.1 

May  1 115.9  99.9  84.6  99.7  80.9 

June  1   123.5  97.6  86.3  102.8  82.7 

July  1    120.8  95.3  84.3  99.0  81.4 

August  1   107.1  98.9  82.7  89.7  77.1 

September  1   . .  95.2  95.8  84.8  85.8  77.1 

October  1   94.6  93.7  88.4  83.4  77.9 

November  1  ...     99.9  90.5  91.5  83.8  77.0 

December  1   . . .  98.6  88.3  87.4  76.0  79.9 

(December  1,  1908,  92.2) 

14.  Using  the  method  of  averages,  derive  an  index  of  sea- 
sonal variations  from  the  following  table  of  farm  prices 
of  wheat  in  the  United  States  (cents  per  bushel)  for  the 
years  1909  to  1918,  inclusive. 


130  INTRODUCTION  TO  ECONOMIC  STATISTICS 


Yearly- 

Monthly 

averages 

; 

averages 

; 

1909 

101.3 

Jan.  1 

109.4 

1910 

96.5 

Feb.  1 

115.2 

1911 

86.9 

March  1 

115.2 

1912 

87.4 

April  1 

116.4 

1913 

78.4 

May  1 

125.6 

1914 

88.4 

June  1 

126.0 

1915 

105.2 

July  1 

117.7 

1916 

125.9 

Aug.  1 

117.9 

1917 

200.8 

Sept.  1 

117.4 

1918 

204.3 

Oct.  1 

116.5 

Nov.  1 

119.7 

Dec.  1 

118.6 

15.  Compute  and  plot  the  cycles  in  the  price  of  wheat,  1909- 
1913,  as  measured  from  a  line  of  least  squares.  Use  the 
data  of  Exercise  13. 

16.  From  financial  journals  and  other  sources  obtain  monthly 
or  weekly  quotations  for  recent  and  current  dates  illus- 
trating barometric  subjects  such  as  are  mentioned  on 
pages  66  and  124.  Plot  the  data  and  construct  trends. 
On  the  basis  of  these  barometers  and  such  other  informa- 
tion as  may  be  available,  make  a  forecast  of  business 
conditions  for  the  immediate  future,  allowing  for  seasonal 
variations. 


CHAPTER  VI 

CORRELATION 

Correlation  Defined.  A  study  of  the  cycles  of  busi- 
ness barometers  leads  to  the  problem  of  classifying 
and  measuring  the  relationships  among  them.  These 
relationships  may  be  discovered  in  various  forms  and 
degrees.  For  example,  the  cycles  of  building  permits 
and  pig  iron  production  will  be  found  to  move  some- 
what closely  together.  Then  again,  stock  prices  and 
commodity  prices  form  similar  waves,  though  the  lat- 
ter usually  follow  a  few  months  behind.  On  the  other 
hand,  commodity  prices  and  business  failures  show 
opposite  movements — when  one  is  up  the  other  is  down. 
All  such  relationships  between  two  sets  of  data  are 
known  as  correlations.  When  the  two  sets  of  cycles 
agree,  the  correlation  is  called  positive ;  when  they  dis- 
agree, it  is  negative.  When  the  cycles  are  not  quite 
coincident  in  point  of  time,  the  one  which  follows  is 
said  to  show  a  **lag"  of  a  given  interval.^    Correla- 

*  The  term  ' '  lag ' '  is  also  sometimes  used  to  designate  a  smaller 
degree  of  variation  occurring  in  one  series  than  in  another  comparable 
to  it.  Thus  the  lag  of  retail  prices  behind  wholesale  prices  ia  largely 
a  matter  of  degree,  and  only  slightly  a  matter  of  time.  But  it  is  the 
time  element  only  that  enters  into  the  calculation  of  correlation.  In 
allowing  for  the  lag,  the  series  coming  later  in  time  is  considered  aa 
moved  back  by  the  length  of  the  lag,  and  the  corresponding  items  are 
then  compared.  When  the  length  of  the  lag  is  difficult  to  determine, 
estimates  must  be  made,  and  the  correlation  computed  on  the  basia  of 
each  estimate.  The  lag  resulting  in  the  most  marked  correlation  is 
assumed  to  be  the  correct  one.  In  determining  the  lag  it  is  often 
necessary  to  take  into  account  the  causal  relation  existing  between  the 
two  series  under  consideration,  as  in  a  case  where  the  assumption  of  a 
lag  for  one  series  results  in  a  positive  correlation,  while  the  transfer 
of  the  lag  to  the  other  series  results  in  a  negative  correlation. 

131 


132  INTRODUCTION  TO  ECONOMIC  STATISTICS 

tion  carries  the  idea  of  a  fundamental  relationship: 
either  one  phenomenon  acts  or  reacts  upon  the  other, 
or  both  are  due  to  common  causes.  The  principle  is 
not  limited  to  time  series.  Comparison  might  be  made, 
for  example,  between  the  advertising  and  the  rate  of 
earnings  of  given  business  firms.  But  in  any  case 
the  principle  would  be  the  same  as  before,  and  the 
methods  used  would  be  practically  identical. 

The  Graphic  Method.  A  fairly  good  study  of  cor- 
relation in  economic  phenomena  can  often  be  made 
mthout  any  more  elaborate  methods  than  those  al- 
ready described  in  isolating  and  plotting  the  cycles. 
Two  time  series,  reduced  to  standard  deviation  cycles 
and  plotted  on  equal  horizontal  scales  may  be  very  well 
compared  by  superimposing  one  on  the  other.  To  fa. 
cilitate  comparison,  one  may  be  drawn  on  a  trans- 
parent medium,  such  as  tracing  cloth ;  or  a  mimeoscope 
may  be  used.  By  shifting  the  superimposed  cycles 
back  and  forth,  the  lag  may  be  fairly  accurately  deter- 
mined. The  correlation  may  be  described  as  posi- 
tive or  negative,  high,  moderate  or  low,  and  the  lag 
and  its  consistency  may  be  stated.  This  is  the  method 
adopted  by  the  Review  of  Economic  Statistics  in  its 
study  of  the  correlations  existing  among  24  business 
barometric  series  for  the  years  1903-1914. 

Method  of  Concurrent  Deviations.  It  is  often  desir- 
able, however,  to  measure  the  degree  of  correlation  in 
precise  mathematical  terms.  This  is  particularly  true 
when  correlated  data  are  being;  used  in  support  of  a 
given  theory.  In  order  to  obtain  a  precise  result, 
mathematical  methods  developed  originally  for  use  in 
biometrics  have  been  borrowed  and  adapted. 


CORRELATION  133 

As  an  introduction  to  the  mathematical  methods  of 
measuring  correlation,  we  may  take  up  a  simple  for- 
mula which  is  well  adapted  to  the  comparison  of  short- 
time  fluctuations.  The  formula  is  the  expression  of 
what  is  called  the  method  of  concurrent  deviations.  It 
may  be  illustrated  by  applying  it  to  a  comparison  be- 
tween the  short-time  fluctuations  of  real  wages  and  per 
capita  production  in  the  United  States  (cf.  pp.  51,  53, 
and  81).  The  fluctuations  may  be  most  readily  deter- 
mined by  reference  to  a  graph  of  each  series  (cf.  Fig. 
6,  p.  101).  If  at  any  given  year  the  line  makes  an  in- 
verted angle,  like  a  caret  (A),  the  fluctuation  is  reg- 
istered on  the  index  as  positive  (  +  ).  If  the  angle 
is  V-shaped,  it  is  registered  as  negative  (-).  If  no 
angle  is  formed,  the  year  is  indicated  as  neutral  (0). 
In  some  cases  it  may  not  be  possible  to  determine  from 
the  graph  whether  the  angle  is  neutral,  or  slightly  posi- 
tive or  negative;  in  which  case  resort  may  be  had  to 
the  data. 

After  the  deviations  of  both  series  have  all  been 
registered,  they  are  compared  across,  item  by  item. 
If  in  a  given  year  both  indexes  show  a  positive  fluctua- 
tion, one  agreement  is  counted.  If  positive  and  nega- 
tive meet,  one  disagreement  is  counted.  If  one  or  both 
of  the  fluctuations  of  a  given  year  are  neutral,  one- 
half  is  added  to  both  the  agreements  and  disagree- 
ments. When  this  summation  is  complete,  the  larger 
of  the  two  totals  thus  obtained  is  designated  as  the 
number  of  concurrent  deviations,  denoted  in  the  for- 
mula by  the  letter  C.  The  sign  of  the  coefficient  to  be 
obtained  by  the  use  of  the  formula  is  determined  by 
the  nature  of  the  concurrences.     If  they  are  agree- 


134  INTRODUCTION  TO  ECONOMIC  STATISTICS 

ments,  the  sign  is  positive ;  if  disagreements,  the  sign 
is  negative.  The  formula  for  correlation  (B)  as  thus 
measured  is : 

In  the  case  of  the  wage  and  production  indexes  just 
mentioned,  the  number  of  disagreements,  or  concur- 
rences, totals  to  33>4  and  the  number  of  comparisons  is 
49.    The  formula  therefore  becomes: 

=  —  .61 

The  derivation  of  the  formula  is  of  little  importance, 
as  it  is  patterned  empirically  on  the  one  next  to  be 
described.  The  significance  of  the  coefficient  will  be- 
come evident  in  the  same  connection. 

The  Pearson  Method.  We  come  now  to  the  so-called 
Pearson  **r",  the  most  satisfactory  method  to  apply 
to  straight  line  correlations:  that  is,  to  those  which 
when  graphed  show  an  approximation  to  a  straight 
line  rather  than  a  curve.  This  statement  does  not  refer 
to  the  trends  of  the  two  series  taken  separately,  but 
only  to  the  trend  formed  by  the  two  sets  of  cycles 
plotted  as  x  and  y,  respectively.  In  explaining  the 
method,  it  is  most  convenient  to  begin  with  two  sets 
of  cycles,  or  deviations,  already  reduced  to  units  con- 
sisting of  their  respective  standard  deviations.  The 
following  table  gives  two  such  series,  and  the  process 
of  finding  their  correlation.  The  two  cycles  are 
graphed  as  coordinates  in  Figure  14,  page  136. 

Since  the  units  used  in  both  cases  are  standard  devia- 
tions, the  spread  on  the  two  axes,  as  measured  by  the 


CORRELATION  135 

CORRELATION   OF   PRICES    (X)    AND    EMPLOYMENT    (Y) 
JANUARY,  1920,  TO  JANUARY,  1921 
Cycles,  in  Units  of  Standard  Deviation 


Totals 


X 

y 

X^ 

xy 

.75 

.59 

.5625 

.4425 

.94 

.85 

.8836 

.7990 

.91 

.51 

.8281 

.4641 

.88 

1.10 

.7744 

.9680 

.89 

.68 

.7921 

.6052 

.57 

.51 

.3249 

.2907 

.37 

.34 

.1369 

.1258 

.18 

.42 

.0324 

.0756 

-.14 

-.09 

.0196 

.0126 

-.53 

-.34 

.2809 

.1802 

-.98 

-.59 

.9604 

.5782 

-1.74 

-1.27 

3.0276 

2.2098 

-2.10 

-2.71 
0 

4.4100 

5.6910 

0 

13.0334 

12.4427 

12.4427 

r  = 

1  o  nnoA      — 

.96 

13.0334 

cycles  squared,  must  necessarily  be  equal.  If  every 
deviation  in  one  series  concurs  with  an  equal  deviation 
in  the  other  series,  the  points  when  plotted  will  neces- 
sarily fall  on  a  diagonal  sloping  upward  from  left  to 
right  at  45°.  If  positive  deviations  concur  with  nega- 
tive, the  points  will  lie  in  a  diagonal  sloping  downward 
from  left  to  right  at  45°.  In  the  first  case  a  line  of  least 
squares  drawn  through  the  points  will  necessarily  have 
a  slope  of  -f  1,  and  in  the  second  case  of  — 1.  These 
are  obviously  the  largest  results,  both  positive,  and 
negative,  that  could  be  obtained  from  two  such  correl- 
ative series.  A  neutral  result  of  zero  would  be  ob- 
tained if  the  points  as  plotted  fall  in  haphazard  posi- 
tions about  the  two  axes.  The  slope  of  the  line  of 
least  squares  (the  tangent  of  its  angle  \vith  the  X-axis) 


136  INTRODUCTION  TO  ECONOMIC  STATISTICS 

is  therefore  taken  as  the  measure  of  correlation.    Its 
basic  formula  is : 

2  X  y 
^""2x2 

In  ordinary  work  it  is,  of  course,  necessary  first  to 
find  a  trend  for  each  series,  if  the  cycles  are  to  be  meas- 
ured. If  the  deviations  are  taken  from  the  average  of 
each  series,  the  general  direction  and  form  of  the  two 
lines  will  be  contrasted.    This  is  equivalent  to  assum- 


^<r 


/<r 


O 


-/a- 


-2<T 


X' 

/ 

At 
A/ 

• 

y 

-^<r         -/<r 


/<r  ^<r 


FiOTjRi;  14.  Correlation  of  price  cycles  (i)  and  employment  cycles  (y), 
expressed  in  units  of  the  standard  deviation  of  each  series,  respectively. 
Line  of  least  squares  (solid  line),  and  line  of  unit  slope  (broken  line). 


CORRELATION  137 

ing  a  horizontal  trend  as  a  basis  of  measurement.  In 
certain  cases  this  comparison  may  be  desired.  But 
usually  it  is  advisable  to  measure  either  the  two  trends 
by  eliminating  the  cycles  or  to  measure  the  cycles  as 
taken  from  the  trend.  The  latter  is  the  usual  pro- 
cedure, since  interest  generally  lies  in  a  comparison 
of  the  cycles. 

The  computing  of  a  coeflficient  of  correlation  from 
data  which  require  the  finding  of  trends  is  illustrated 
in  Table  XVIII.  The  work  is  for  the  most  part  self- 
explanatory,  since  the  trends  are  found  by  processes 
already  explained.  Instead,  however,  of  using  the 
standard  deviations  as  the  units  in  which  to  express 
the  cycles,  the  original  units  are  employed  throughout. 
The  reduction  to  standard  deviation  units  is  in  effect 
obtained  by  inserting  a^  and  fTg  in  the  denominator  of 
the  formula  for  r.  But  a  substitute  must  be  found  for 
2x-,  which  in  the  formula  as  just  used  was  also  in  terms 
of  the  standard  deviation.  The  required  substitute  is 
N.    This  may  be  seen  by  recalling  that  in  the  standard 

deviation  series  /l/  -^    ==  a  =  1.    Hence  ^x^  must 

equal  N.  The  formula  for  the  line  of  least  squares  as 
applied  to  the  previous  problem  in  correlation  may 
therefore  be  transformed  into  the  Pearson  correlation 
formula,  thus : 

„  (both  X  and  y  being  expressed  in 

r  =  -y-^       units  of  their  respective  standard 
deviations.) 
__    Sxy      (in  which  x  and'i/  are  expressed  in 
Njidg     terms  of  the  original  units.) 
_        ^xy         (an  alternate  form  obtained  by  sub- 
V2x22y2      stitution.) 


138  INTRODUCTION  TO  ECONOMIC  STATISTICS 


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I   1   I   I  > 


COCMrHOr-ICNCOTtlkn 


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b-^  CQ  in  CO  00  in  CO  in  oo  ti<  •>*' 

rHr-li— li-(Csll-Hi— li-liHrHi— i 


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CORRELATION 


139 


2  3  u 
s  a  ^ 


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f  O  C<i  T-H  rH  CO       *     l'  «?      f      f 


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t~  «o  CO  ai  <M 


N  00  «D  t-;  i-j  c>  00  cci  iq  ■*  iq 

l'  N  N  CO  CO  t--       *  r-J  CO  r-H  rH 


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140  INTRODUCTION  TO  ECONOMIC  STATISTICS 

The  Probable  Error.  It  will  be  seen  that  the  first 
part  of  Table  XVIII  records  merely  the  finding  of  the 
trends  (lines  of  least  squares)  for  the  two  series.  The 
figures  have  not  been  carried  out  beyond  approximate- 
ly one  per  cent  accuracy.  The  production  and  price 
cycles  shown  in  the  latter  part  of  the  table  are  obtained 
by  the  usual  method  of  subtracting  the  trend  items 


'Z<T, 


Figure  15.  Variations  in  production  and  price  of  pig  iron,  United 
States,  1908.  Standard  deviation  units,  measured  from  the  line  of 
least  squares  (r  =  .68). 

from  the  corresponding  items  in  the  original  series. 
The  standard  deviation  of  each  of  the  resulting  series 
of  cycles  and  the  summation  of  the  xy  's  are  next  found. 
The  Pearson  formula  is  then  applied,  and  the  result, 
r  =  .45,  is  obtained.  To  this  result  is  appended  what 
is  known  as  the  ''probable  error"  (P.E.),  the  word 
** error"  being  here  used  merely  in  the  sense  of  di- 
vergence, as  in  the  theory  of  least  squares.  The  for- 
mula for  the  probable    error    expresses  the  quartile 


CORRELATION  141 

deviation  from  the  coefficient  of  .45  which  would  nor- 
mally appear  by  the  operation  of  the  laws  of  chance. 
The  probable  error  therefore  gives  some  idea  of  the 
range  over  which  the  value  of  r  has  an  even  chance 
of  deviating,  and  may  be  used  in  estimating  the  sig- 
nificance of  the  correlation.  On  the  basis  of  experience 
it  is  assumed  that  if  the  value  of  r  is  as  low  as  thirty, 
or  if  the  probable  error  is  as  high  as  one-third  of  r, 


Feb.   Mar.  Apr.  May  Jan.  Jul.    Aiy.  Sep.   Oct   Nw.  Dec 


Figure  16.  Seasonal  variations  in  the  visible  supply  and  the  price 
of  wheat,  United  States,  1909-1913  (r  =  — .87,  prices  preceding  one 
month).     For  data,  see  Exercise  8,  page  150. 

correlation  is  barely  indicated.  If,  however,  r  is  as 
high  as  fifty,  and  if  the  probable  error  is  not  more 
than  one-fifth  of  r,  correlation  is  clearly  indicated. 
Between  these  limits  a  correlation  may  be  regarded 
as  more  or  less  tentatively  indicated. 

The  Relation  of  Output  to  Price.  The  result 
obtained  in  Table  XVIII  indicates  that  in  the  iron  in- 
dustry production  is  directly  adjusted  to  meet  demand 
as  reflected  in  the  price.    When  the  price  is  high,  pro- 


142  INTRODUCTION  TO  ECONOMIC  STATISTICS 

duction  therefore  is  high,  and  vice  versa.  This  relation 
is  seen  to  be  very  close  in  certain  years  if  monthly 
data  are  used  (cf.  Figure  15).  The  case  is  doubtless 
t}T)ical  of  a  large  part  of  manufacturing,  particularly 
when  the  process  is  relatively  short.  But  in  agricul- 
ture, where  the  maturing  of  the  output  is  determined 
by  the  seasons,  contemporaneous  movements  of  out- 
put and  prices  are  negatively  related.  This  fact  is 
shown  indirectly  by  Figure  16.  About  75  7^  of  the 
world 's  wheat  crop  is  harvested  in  the  months  of  June, 
July,  and  August,  with  a  consequent  rapid  depression 


Figure  17.  Comparison  of  corrected  figures  for  index  of  physical 
production  of  crops  and  index  of  crop  prices.  The  long  tine  movements, 
or  secular  trends,  have  been  eliminated,  and  the  two  series  have  been 
expressed  in  comparable  units. 

Reproduced  from  the  Eeview  of  Economic  Statistics,  by  permission 
of  the  editors. 

of  the  price.  Extremely  large  crops  are  normally  fol- 
lowed for  several  months  by  unusually  low  prices,  and 
vice  versa.  This  fact  is  depicted  in  Figure  17,  where 
agricultural  x^roduction  is  plotted  against  the  subse- 
quent price,  rather  than  the  average  price  for  the 
year.  A  comparison  of  the  cycles  of  crops  and  of 
manufactures  suggests  that  the  former  exert  some  pos- 


CORRELATION 


143 


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C2  CO               iH 

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<»  to  00  o  o  O  C-) 
I— 1  CO  (M  I— 1          1— 1  I— 1 

rH 

oi 

II  II            I 

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tH 

tH 

I— 1  t— 1  r-l          iH      1 

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to 

m  CO 

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CO  •* 

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II II 

o       ^-     • 

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MW 

II     II'PII 

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t>    I 


i 


144  INTRODUCTION  TO  ECONOMIC  STATISTICS 

itive  influence  upon  the  latter  within  a  period  of  a  year 
or  two,  but  the  relation  is  not  very  regular.^ 

Correlation  from  Frequency  Tables.  A  somewhat 
diflScult  application  of  the  Pearson  method  of  meas- 
uring correlation  is  encountered  when  the  two  series 
which  are  to  be  compared  are  each  compiled  in  fre- 
quency tables.  The  case  is  illustrated  in  Table  XIX, 
where  average  entrance  examination  grades  and  aver- 
age scholarship  grades  for  the  four  college  years  are 
compared.  The  entrance  examination  data  are  ex- 
pressed in  per  cents,  tabulated  to  the  nearest  multiple 
of  five.  The  scholarship  grades  were  expressed  pri- 
marily in  six  groups,  ranking  downward  in  order  from 
the  first  to  the  sixth.  The  averaging  of  such  groups 
for  four  years  gave  results  carried  out  to  fourths  of  a 
group,  as  shown  in  the  table.  For  convenience  of  cal- 
culation the  values  of  both  scales  are  converted  into 
unit  intervals  measuring  the  deviations,  the  new  scales 
centering  at  a  zero  set  opposite  the  values  (70  and 
314)  which  are  assumed  as  the  averages.  The  number 
of  frequencies  for  the  combined  series  is  written  at 
the  appropriate  coordinate  points  in  the  body  of  the 
table,  while  the  frequencies  for  each  scale  taken  inde- 
pendently are  written  to  the  right  and  below  (column 
F  and  row  F). 

The  initial  steps  in  the  computation  will  be  readily 
understood  by  reference  to  the  short-cut  method  of 
finding  the  standard  deviation.  By  this  method  the 
standard  deviations  for  both  series,  respectively,  are 
determined.    The  finding  of  2xy  is  a  somewhat  long 

*  H.  L.  Mooro,  in  Economic  Cycles,  finds  a  positive  correlation  be- 
tween an  eight-year  crop  cycle  and  general  prices,  allowing  a  four- 
yeaj  lag  to  the  latter. 


CORRELATION  145 

process,  since  each  of  the  frequencies  at  the  coordinate 
points  in  the  body  of  the  table  must  be  taken  into  ac- 
count. Each  of  these  frequencies  is  multiplied  by  its 
two  coordinate  values,  and  the  products  are  totaled. 
To  illustrate,  the  first  three  columns  of  frequencies 
give  the  following  results: 


F 

X 

y 

Fxy 

1 

-6 

0 

0 

1 

-6 

-2 

12 

2 

-5 

-1 

10 

1 

^ 

1 

-4 

2 

-4 

-1 

8 

By  continuing  this  computation,  a  total  result  of  148, 
as  the  value  of  2xy,  is  obtained. 

Since  the  two  inserted  scales  measuring  the  devia- 
tions are  not  centered  with  precision  at  the  two  axes 
of  the  table,  as  determined  by  the  averages  of  the  two 
series,  respectively,  the  plus  and  minus  deviations  from 
the  assumed  averages  will  not  exactly  balance.  Hence 
a  correction  must  be  made  in  the  2xy,  just  as  in  the 
two  standard  deviations.  The  corrections  (Ki  and 
K2)  applied  to  the  finding  of  the  standard  deviations 
are,  of  course,  merely  2FD  -^  N.  It  may  be  shown 
that  Sxy  will  be  increased  by  the  product  of  the  two 
corrections,  for  every  item  included.  The  corrected 
summation  of  the  moments  about  the  coordinate  axes 
is  therefore  expressed : 

2xy  -  NK.Ks 
In  other  respects  the  formula  is  as  previously  used. 
For  convenience,  however,  it  is  written  in  the  revised 
form  shown  at  the  foot  of  the  table. ^     Applying  the 

*  In  correlations  where  the  deviations  to  be  contrasted  are  intended 
to  be  taken  from  the   average  of  each  series,   the  coefficient   may  be 


146  INTRODUCTION  TO  ECONOMIC  STATISTICS 


formula  to  the  data  in  question,  the  value  of  r  is  found 
to  be  .62  ±  .06,  a  well-marked  correlation. 

TABLE  XX 

COERELATION    OF   RANKING    OF   STATES    IN   MANUFACTUR- 
ING AND  IN  LITERACY,   U.  S.,   1860 


STATE 


Alabama 

Arkansas 

Connecticut 

Delaware 

Florida 

Georgia 

Illinois 

Indiana 

Iowa 

Kentucky 

Louisiana 

Maine 

Maryland 

Massachusetts.  . . 

Michigan 

Mississippi 

Missouri 

New  Hampshire. 

New  Jersey 

New  York 

JSTorth  Carolina. . 

Ohio 

Pennsylvania.  . . . 
Rhode  Island.  .  . 
South  Carolina. . 

Tennessee 

Vermont 

Virginia 

Wisconsin 


lUNK   IN 
MANUFAC- 
TURES 
(CAPITAL 
PER  SQUARE 
MILE) 


24 
29 

3 

8 

28 

23 

16 

14 

26 

15 

25 

12 

9 

2 

17 

27 

19 

7 

4 

6 

22 

10 

5 

1 

21 

18 

11 

13 

20 


RANK   IN 

LITERACY 

OF    NATIVE 

WHITES 


23 
24 

2 
21 
22 
27 
14 
17 
13 
25 
18 

5 
15 

1 

9 
16 
19 

6 
11 

7 
29 
12 
10 

8 
20 
28 

3 
26 

4 


1 
5 
1 

13 
6 
4 
2 
3 

13 

10 
7 
7 
6 
1 
8 

11 
0 
1 
7 
1 
7 
2 
5 
7 
1 

10 
8 

13 

16 


D" 


1 

25 

1 

169 

36 

16 

4 

9 

169 

100 

49 

49 

36 

1 

64 

121 

0 

1 

49 

1 

49 

4 

25 

49 

1 

100 

64 

169 

256 


r  =  1  - 


62D» 


P.E. 


.10 


NCN^'-l) 


=:  1  - 


6X1618 


29x840 


.60 


1618 


found  directly  from  the  original  items.  This  is  done  by  the  use  of 
the  formula  given  in  Table  XIX.  An  average  of  zero  is  assumed  for 
both  series,  and  the  items  are  treated  as  positive  deviations  from  this 
average.  The  standard  deviations  are  found  by  the  modified  formula 
explained  on  page  41. 


CORRELATION  147 

The  Method  of  Rank-differences.  One  further  modi- 
fication of  the  Pearson  method  of  correlation,  known 
as  the  method  of  rank-differences,  may  be  noted.  This 
method  has  the  advantage  of  simphcity,  and  is  espe- 
cially applicable  to  comparisons  which  are  made  on  the 
basis  of  approximate  measurements  only.  In  Table 
XX  this  method  is  illustrated  by  applying  it  to  a  com- 
parison of  the  ranking  of  twenty-nine  states  in  1860 
for  manufacturing  and  literacy.  The  rankings  as  here 
shown  are  based  upon  the  census  of  1860.  In  arrang- 
ing such  rankings,  ties  may  sometimes  occur.  In  such 
a  case  the  average  rank  of  the  tied  items  is  applied  to 
each  of  the  items.  Thus  if  the  second  and  third  items 
happen  to  be  equal,  each  is  ranked  2i/^ ;  if  the  second, 
third,  and  fourth  are  equal,  each  is  ranked  3.  When 
the  rankings  have  been  tabulated,  as  shown,  the  dif- 
ference between  the  two  ranks  for  each  state  is  found. 
These  differences  are  then  squared,  and  the  squares 
totaled.  The  formula,  as  given  at  the  foot  of  the  table, 
is  an  adaptation  from  the  one  last  discussed.  Apply- 
ing the  formula,  we  find  that  a  correlation  of  .60 
exists  between  the  two  series.  This  comparison  is 
an  illustration  of  a  number  of  interesting  relationships 
which  may  be  shown  to  exist  between  the  economic 
and  the  social  environment. 

Conclusion.  The  purpose  of  this  chapter  will  have 
been  served  if  the  student  has  gained  a  knowledge  of 
the  simpler  methods  commonly  employed  in  measuring 
correlation.  The  full  theory  of  the  subject  is  very 
complex,  and  is  hardly  within  the  scope  of  an  introduc- 
tory course.  A  caution  must  be  sounded,  however, 
against  an  undiscriminating  application  of  the  meth- 


148  INTRODUCTION  TO  ECONOMIC  STATISTICS 

ods  here  explained.  In  particular,  conclusions  stating 
causal  relationships  should  never  be  based  on  mathe- 
matical processes  alone.  The  data,  their  methods  of 
collection,  and  the  concrete  realities  they  are  assumed 
to  measure,  must  all  be  subjected  to  careful  scrutiny. 
The  same  caution  may  indeed  very  properly  be  ex- 
tended to  the  whole  field  of  statistical  methods.  These 
methods  should  prove  to  be  valuable  tools  in  the  inter- 
pretation of  physical,  biological,  and  social  phenomena, 
but  they  may  be  a  source  of  positive  error  if  their  use 
is  not  directed  by  an  adequate  comprehension  of  the 
field  of  knowledge  in  which  they  are  employed. 

REFERENCES 

Bowley,  Arthur  L.,  Elements  of  Statistics  (4th  Edition),  Part 
II,  Chapters  VI-IX. 

Jevons,  W.  Stanley,  The  Principles  of  Science. 

King,  W.  I.,  "The  Correlation  of  Historic  Economic  Vari- 
ables," Quarterly  Publications  of  the  American  Statistical 
Association,  December,  1917,  pp.  847-853. 

Persons,  W.  W.,  "The  Correlation  of  Economic  Statistics," 
Quarterly  Publications  of  the  American  Statistical  Asso- 
ciation, December,  1910,  pp.  287-322. 

Secrist,  Horace,  Readings  and  Problems  in  Statistical  Meth- 
ods, Chapter  X. 

West,  Carl  S.,  Introduction  to  Mathematical  Statistics. 

Yule,  G.  U.,  An  Introduction  to  the  Theory  of  Statistics, 
Chapters  IX-XII. 

EXERCISES 

1.  On  separate  sheets  of  cross-section  paper  having  the  same 
horizontal  scale,  and!  with  the  vertical  scales  so  ad- 
justed as  to  bring  the  deviations  as  nearly  as  possible 
to  the  same  measured  average,  plot  the  cycles  of  pro- 
duction and  price  as  obtained  in  exercises  1  and  2  of 
the  preceding  chapter.  Similarly  plot  the  cycles  in  the 
interest  rate,  measuring  them  from  Figure  12,  and  the 


CORRELATION  149 

price  cycles  as  shown  in  Figure  6  (pp.  118  and  101). 
Copy  these  cycles  on  tracing  paper.  Describe  the  correla- 
tion of  production  and  price  of  the  two  crops  (allowing  a 
lag  of  one  year  for  prices),  and  of  the  interest  and  price 
cycles. 

2.  By  the  method  of  concurrent  deviations,  measure  the 
following  correlations  (Tables  X,  XI,  and  XV,  pp.  51,  53, 
and  81)  :  (a)  Wholesale  prices  and  per  capita  produc- 
tion (both  concurrently,  and  allowing  a  lag  of  one  year 
for  prices),  and  (b)  Wholesale  prices  and  real  wages. 

3.  Using  the  table  given  in  exercise  8  of  the  preceding  chap- 
ter, and  the  lines  of  least  squares  there  obtained,  measure 
by  the  Pearson  "r"  the  correlation  of  production  and 
price  of  iron  for  each  year  there  studied. 

4.  From  the  data  on  page  53,  find  the  correlation  between 
wages  and  the  cost  of  living  for  the  years  1913-1920 
inclusive  (Pearson  "r").  Measure  the  deviations  from 
the  average  of  each  series,  respectively;  that  is,  assume 
a  horizontal  trend, 

5.  Reduce  the  deviations  obtained  in  the  preceding  exer- 
cise to  units  of  the  standard  deviation  of  each  series, 
respectively,  and  plot  as  coordinates  the  two  sets  of 
deviations  thus  obtained.  Compute  the  line  of  least 
squares  for  the  points  so  plotted,  and  show  that  the 
slope  of  this  line  is  identical  with  the  coefficient  of  cor- 
relation. 

6.  Correlate  the  following  indexes  (Pearson  "r")  taking 
the  deviations  from  the  average,  without  finding  a  trend. 

Explain  the  significance  of  the  result. 

Year  Prices         Unemployment 

1912 110  70 

1913 100  120 

1914 90  140 

1915 90  100 

1916 110  70 

7.  Find  the  Pearson  coefficient  of  correlation  for  the  in- 

dexes of  normal  seasonal  variations  of  merchandise  ex- 
ports from  the  United  States  and  the  price  of  sterling 
exchange  at  New  York,  as  follows: 


150  INTRODUCTION  TO  ECONOMIC  STATISTICS 

Month                                         Exports  Sterling 

January  110  100 

February   95  108 

March    99  109 

April    90  115 

May   87  116 

June   80  120 

July    78  119 

August   85  106 

September   98  74 

October   125  70 

November    123  80 

December    130  83 


(a)  Find  the  Pearson  coefficient  of  correlation  measur- 
ing the  relationship  between  the  following  indexes  of 

seasonal  variation  in  the  visible  supply  and  the  price  of 
wheat  in  the  United  States,  based  on  the  years  1909-1913. 

(b)  Find  the  coefficient,  as  before,  but  assuming  that 
prices  tend  to  anticipate  the  supply  by  about  a  month. 

Visible  Price 

Month  supply  (first  of  mo.) 

January   139  97 

February    130  100 

March  122  101 

April    112  101 

May    89  103 

June   69  106 

July    52  104 

August    60  100 

September   77  97 

October    99  97 

November 118  98 

December 133  96 


The  following  correlation  table  presents  entrance  groups 
(vertical  scale)  and  scholarship  groups  (horizontal 
scale)  for  a  certain  class  of  students.  Find  the  coef- 
ficient of  correlation  (Pearson  "r")  and  the  probable 
error. 


CORRELATION 


151 


10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

2 

1 

1 
1 

1 

2 
1 

1 

1 

4 

2 

3 

1 

1 

1 

1 

2 

1 

1 

1 

1 

1 

3 

2 

1 

2 

10.  The  following  correlation  table  classifies  to  the  nearest 
twenty-five  per  cent  sixty-nine  important  commodities 
according  to  their  price  indexes  (base,  1913)  in  May, 
1920,  and  May,  1921  {Monthly  Labor  Review,  Aug., 
1921,  pp.  84-85).  The  correlation  measures  approxi- 
mately the  evenness  of  the  price  changes  occurring 
between  the  two  dates.    Find  Pearson's  "r." 


400 
^  375 
2  350 
.325 
^  300 
§  275 
g-250 

^  200 
M  175 
flj  150 
•i  125 
Ph  100 
75 


Price  Indexes,  May,  1920 


oicoiooicoiooiooiooinoiooiooinoiooiooio 
oocimt^ocMiot^ooainii^oiMkot^oocimi^-oiMiot^ooa 

1 


113 


1  2 


1  1 


1248859444545111 


1    69 


11.  The  following  table  shows  the  ranking  of  states  in 
(a)  Noted  men  bom  in  state,  per  1000  population  in 
1880,  (b)  Population  per  square  mile  in  1890,  and 
(c)  Per  cent  of  urban  population  in  1890.  By  the 
method  of  rank-differences  measure  the  correlations 
existing  among  these  three  series. 


(a) 

Alabama    23 

Arkansas    29 


(b) 

(c) 

24 

25.5 

28 

28 

152  INTRODUCTION  TO  ECONOMIC  STATISTICS 

Connecticut    4 

Delaware    8 

Florida    28 

Georgia   26 

Illinois    16 

Indiana  15 

Iowa   19 

Kentucky    18 

Louisiana  25 

Maine   5 

JMaryland  10 

Massachusetts    2 

Michigan    17 

Mississippi  27 

Missouri   21 

New  Hampshire  3 

New  Jersey  11 

New  York   7 

North   Carolina    22 

Ohio    9 

Pennsylvania    12 

Rhode  Island 6 

South  Carolina 20 

Tennessee    24 

Vermont    1 

Virginia   13 

Wisconsin 14: 


4 

3 

9 

11 

29 

20 

22 

23 

10 

10 

11 

17 

20 

19 

12 

21 

26 

18 

27 

9 

7 

8 

2 

2 

18.5 

14 

25 

29 

16 

16 

14 

6 

3 

5 

5 

4 

21 

27 

8 

12 

6 

7 

1 

1 

17 

25.5 

13 

24 

18.5 

13 

15 

22 

23 

15 

APPENDIX  I 

LABORATORY  MATERIAL  AND  REFERENCES 

Equipment  for  Graphic  Work.  While  the  larger  part  of 
statistical  work  may  be  done  without  any  systematic  train- 
ing in  mechanical  drawing,  yet  some  degree  of  skill  in  this 
field  is  necessary  if  graphic  representations  are  to  be  satis- 
factorily prepared.  The  necessary  degree  of  skill  may  readily 
be  acquired.  The  student  should  provide  himself  with  a 
drawing  board,  celluloid  triangles  and  irregular  curves,  a 
ruler  with  decimal  subdivisions  of  the  inch,  a  ruling  pen, 
some  round-pointed  and  fine  pens  for  lettering,  India  ink, 
several  styles  of  cross-section  paper,  and  a  loose-leaf  note- 
book. If  he  is  unfamiliar  with  the  use  of  drafting  materials, 
he  should  read  the  introductory  directions  given  in  an  ele- 
mentary treatise  on  mechanical  drawing. 

In  addition  to  the  material  just  mentioned,  some  of  the 
more  complicated  apparatus  used  in  an  engineer's  drafting 
room  will  be  found  useful.  This  equipment  may  include  a 
drafting  machine,  a  pantagraph,  a  line-spacer,  a  map-meas- 
urer, and  a  planimeter.  A  blue-print  outfit  is  also  very  use- 
ful, indeed  is  almost  a  necessity,  unless  some  improved  copy- 
ing device  like  the  photostat  is  available  for  use.  For  ele- 
mentary work  a  simple  S^xlO''  photography  printing  frame, 
and  corresponding  blue-print  paper,  will  be  found  quite  sat- 
isfactory and  inexpensive. 

Lettering.  It  is  not  difficult  to  learn  to  draw  freehand  the 
italic  letters  used  by  draftsmen.  Directions  for  such  work  will 
be  found  in  "Lettering  for  Draftsmen,  Engineers,  and  Stu- 
dents," by  Charles  W.  Reinhardt  (D.  Van  Nostrand  Com- 
pany, New  York),  or  in  other  books  on  the  same  subject. 

Other  Material.  There  are  several  aids  to  statistical  work 
that  will  materially  lighten  the  drudgery  incidental  to  long 
mathematical  processes.  The  most  common  of  these  is  the 
slide  rule.  A  ten-inch  rule,  giving  squares  and  cubes,  will 
be  found  sufficient  for  the  greater  part  of  the  work  involving 
multiplication,  division,  and  powers  or  roots.    The  slide  rule 

153 


154  INTRODUCTION  TO  ECONOMIC  STATISTICS 

is  not  difficult  to  use,  and  should  be  mastered  by  every  stu- 
dent of  statistics.  Besides  being  an  inexpensive  and  portable 
device  for  mathematical  operations,  it  will  be  found  useful 
in  the  drawing  of  logarithmic  or  ratio  graphs,  which  are  now 
coming  into  general  use.  If,  however,  it  is  necessary  to  ob- 
tain products  or  quotients  accurate  to  four  or  five  significant 
figures,  a  large  cylindrical  slide  rule  may  be  used,  such  as  the 
Thatcher,  though  this  is  less  convenient  and  considerably 
more  expensive.  For  powers,  roots,  and  reciprocals,  elaborate 
printed  tables  are  obtainable.  If  possible,  an  adding  machine 
(listing)  should  be  available  for  occasional  use,  such  as  the 
Dalton,  the  Burroughs,  or  the  Federal.  While  such  a  machine 
is  a  convenience,  a  calculating  machine  is  an  absolute  neces- 
sity if  very  extensive  work  is  to  be  attempted;  and  it  is  well 
for  the  student  to  become  acquainted  with  its  operation.  Sev- 
eral successful  models  are  now  on  the  market,  among  which 
may  be  mentioned  the  Burroughs,  the  Comptometer,  the  Mon- 
roe, and  the  Marchant.  The  last  two  are  dial  machines,  par- 
ticularly adapted  for  subtractions  and  divisions.  For  certain 
kinds  of  statistical  work  tabulating  machines  (Hollerith  and 
Powers  types)  are  required,  but  these  machines  are  so  com- 
plex and  expensive  that  they  can  hardly  be  made  available 
except  in  the  larger  laboratories. 

Recording.  Laboratory  exercises  and  other  statistical  work 
should  be  recorded  fully,  and  should  be  put  in  clear  and  neat 
form.  Every  graph  should  be  accurately  labeled,  and  the 
units  used  in  each  scale  should  be  indicated.  When  the  scales 
of  a  graph  do  not  begin  at  the  zero  point,  the  initial  coor- 
dinates should  not  be  drawn  more  heavily  than  the  others, 
since  they  are  likely  to  be  looked  upon  as  base  lines  if  so 
drawn.  In  so  far  as  is  practicable,  the  tables  of  data  from 
which  a  graph  is  drawn  should  accompany  the  figure,  and 
the  source  should  be  noted.  Graphing  and  lettering  should 
be  done  in  pencil  first;  the  pencil  draft  may  then  be  com- 
pleted wnth  India  ink,  and  the  pencil  lines  erased  with  a  soft 
rubber.  Errors  in  calculation  should  not  be  tolerated.  All 
mathematical  operations  should  be  performed  twice,  or  some 
other  reliable  method  of  checking  should  be  adopted.  Data 
copied  from  an  original  source  should  always  be  carefully 
verified.  Tables  and  mathematical  processes  may  most  con- 
veniently be  recorded  on  cross-section  paper  having  one-fifth 
or  one-sixth  inch  spacing.  In  any  given  study  the  conclu- 
sions should  be  brought  out  clearly,  and  their  significance  ex- 
plained. 

Various  Types  of  Graphs.    Most  of  the  types  of  graphs  in 


APPENDIX  I  155 

common  use  have  been  illustrated  in  the  preceding  pages. 
In  addition,  mention  may  be  made  of  certain  elementary- 
types.  One  of  these  is  the  bar  diagram,  in  which  bars  of 
uniform  width,  and  proportional  in  length  to  given  magni- 
tudes, are  used.  They  are  drawn  horizontally,  except  in  time 
series.  When  they  are  subdivided,  the  parts  are  distinguished 
by  various  kinds  of  cross-hatching  and  shading.  Another  is 
the  "pie  diagram,"  in  which  a  circle  is  subdivided  by  radii. 
This  diagram  is  particularly  adapted  to  the  representation  of 
percentage  subdivisions,  such  as  the  relative  expenditures  for 
certain  classes  of  goods  in  a  family  budget.  Another  type  is 
the  polar  chart,  designed  for  graphing  seasonal  data.  Draw- 
ings of  similar  surfaces  and  solids  are  sometimes  used  in 
the  representation  of  given  magnitudes.  It  should  be  re- 
membered that,  geometrically,  magnitudes  compared  by  the 
use  of  similar  surfaces  vary  as  the  square  of  the  dimensions; 
and  by  the  use  of  similar  solids,  as  the  cube  of  the  dimensions. 
Sometimes  in  such  drawings  it  is  explicitly  stated  that  the 
ratio  is  represented  by  one  dimension  only,  as  when  the 
military  forces  of  different  countries  are  set  forth  by  means 
of  drawings  of  soldiers  whose  heights  are  proportional  to  the 
size  of  the  armies.  Such  drawings  are  not  scientific,  how- 
ever, and  are  justified  only  in  material  of  a  very  popular 
nature.  In  general,  the  use  of  similar  surfaces  and  solids 
in  the  representation  of  magnitudes  is  to  be  avoided.  A  more 
complex  type  of  graph  is  the  statistical  map.  This  may  be 
drawn  in  so  many  different  ways  that  a  general  description  is 
impossible.  The  student  having  occasion  to  use  it  should  con- 
sult the  excellent  examples  contained  in  the  Statistical  Atlas 
of  the  United  States. 

Sources,  References,  and  Tables.  Brief  summaries  are 
given  below  of  the  principal  sources  of  statistical  material, 
and  the  textbook  references  and  statistical  tables  which  are 
most  likely  to  be  of  use  in  connection  with  an  introductory 
course. 

SOURCES  OF  STATISTICAL  DATA 

Aldrich  Report  {Senate  Report  No:  1394) 

Annalist 

Bradstreet's 

Commercial  and  Financial  Chronicle 

Dun's  Review 

Federal  Reserve  Bidletin 

Financial  Review  {year  hook) 


156  INTRODUCTION  TO  ECONOMIC  STATISTICS 

Monthly  Labour  Review 

Monthly  Review  (Federal  Reserve  Bank  of  New  York) 
Monthly  Summary  of  Foreign  Commerce  of  the  United  States 
Review  of  Economic  Statisitics  [Harvard) 
Statesman's  Yearbook 

Statistical  Abstract  of  the  United  States  (yearbook) 
Statistical  services:  Bahs&n's,  Banker's  Statistical  Corpora- 
tion, Brookmire's,  and  Prentice-Hall. 
Survey  of  Current  Business. 
United  States  Census 

Weather,  Crops,  and  Markets  (U.  S.  Dept.  of  Agriculture) 
World  Almanac 
Yearbook  of  the  Department  of  Agriculture 

TEXTBOOKS,  TABLES,  AND  GENERAL 
REFERENCES 

American  Econmnic  Review  (Bi-monthly) 
Bailey  and  Cummings,  Statistics 
Barker,  E.  H.,  Computing  Tables  and  Formulas 
Barlow's  Tables 

Bowley,  A.  L.,  Elements  of  Statistics 
Brinton,  W.  C.,  Graphic  Methods  for  Presenting  Facts 
Copeland,  M.  T.,  Business  Statistics 
Davenport,  C.  B.,  Statistical  Methods 
Jordan,  D.  F.,  Business  Forecasting 
Journal  of  Political  Economy  (Monthly) 
King,  W.  I.,  Elements  of  Statistical  Method 
Marshall,  Wm.  C,  Graphical  Methods 
Quarterly  Journal  of  Economics 

Quarterly  Publications  of  the  American  Statistical  Associa- 
tion 
Secrist,  H.,  An  Introduction  to  Statistical  Methods 
Secrist,  H.,  Readings  and  Problems  in  Statistical  Methods 
West,  C.  S.,  Introduction  to  Mathematical  Statistics 
Whipple,  G.  C,  Vital  Statistics 


APPENDIX  II 

TABLE  OF  POWERS  AND  ROOTS 

Square 

Cube 

No. 

Square 

Cube 

Root 

Root 

1 

1 

1 

1.000 

1.000 

2 

4 

8 

1.414 

1.259 

3 

9 

27 

1.732 

1.442 

4 

16 

64 

2.000 

1.587 

6 

25 

125 

2.236 

1.709 

6 

36 

216 

2.449 

1.817 

7 

49 

343 

2.645 

1.912 

8 

64 

512 

2.828 

2.000 

9 

81 

729 

3.000 

2.080 

10 

100 

1,000 

3.162 

2.154 

11 

121 

1,331 

3.316 

2.223 

12 

144 

1,728 

3.464 

2.289 

13 

169 

2,197 

3.605 

2.351 

14 

196 

2,744 

3.741 

2.410 

15 

225 

3,375 

3.872 

2.466 

16 

256 

4,096 

4.000 

2.519 

17 

289 

4,913 

4.123 

2.571 

18 

324 

5,832 

4.242 

2.620 

19 

361 

6,859 

4.358 

2.668 

20 

400 

8,000 

4.472 

2.714 

21 

441 

9,261 

4.582 

2.758 

22 

484 

10,648 

4.690 

2.802 

23 

529 

12,167 

4.795 

2.843 

24 

576 

13,824 

4.898 

2.884 

25 

625 

15,625 

5.000 

2.924 

26 

676 

17,576 

5.099 

2.962 

27 

729 

19,683 

5.196 

3.000 

28 

784 

21,952 

5.291 

3.036 

29 

841 

24,389 

5.385 

3.072 

30 

900 

27,000 

5.477 

3.107 

31 

961 

29,791 

5.567 

3.141 

32 

1,024 

32,768 

5.656 

3.174 

33 

1,089 

35,937 

5.744 

3.207 

34 

1,156 

39,304 

5.830 

3.239 

36 

1,225 

42,875 
157 

5.916 

3.271 

158  INTRODUCTION  TO  ECONOMIC  STATISTICS 


Square 

Cube 

No. 

Square 

Cube 

Root 

Root 

36 

1,296 

46,656 

6.000 

3.301 

37 

1,369 

50,653 

6.082 

3.332 

38 

1,444 

54,872 

6.164 

3.361 

39 

1,521 

59,319 

6.244 

3.391 

40 

1,600 

64,000 

6.324 

3.419 

41 

1,681 

68,921 

6.403 

3.448 

42 

1,764 

74,088 

6.480 

3.476 

43 

1,849 

79,507 

6.557 

3.503 

44 

1,936 

85,184 

6.633 

3.530 

45 

2,025 

91,125 

6.708 

3.556 

46 

2,116 

97,336 

6.782 

3.583 

47 

2,209 

103,823 

6.855 

3.608 

48 

2,304 

110,592 

6.928 

3.634 

49 

2,401 

117,649 

7.000 

3.659 

50 

2,500 

125,000 

7.071 

3.684 

51 

2,601 

132,651 

7.141 

3.708 

52 

2,704 

140,608 

7.211 

3.732 

53 

2,809 

148,877 

7.280 

3.756 

54 

2,916 

157,464 

7.348 

3.779 

55 

3,025 

166,375 

7.416 

3.802 

56 

3,136 

175,616 

7.483 

3.825 

57 

3,249 

185,193 

7.549 

3.848 

58 

3,364 

195,112 

7.615 

3.870 

59 

3,481 

205,379 

7.681 

3.892 

60 

3,600 

216,000 

7.745 

3.914 

61 

3,721 

226,981 

7.810 

3.936 

62 

3,844 

238,328 

7.874 

3.957 

63 

3,969 

250,047 

7.937 

3.979 

64 

4,096 

262,144 

8.000 

4.000 

65 

4,225 

274,625 

8.062 

4.020 

66 

4,356 

287,496 

8.124 

4.041 

67 

4,489 

300,763 

8.185 

4.061 

68 

4,624 

314,432 

8.246 

4.081 

69 

4,761 

328,509 

8.306 

4.101 

70 

4,900 

343,000 

8.366 

4.121 

71 

5,041 

357.911 

8.426 

4.140 

72; 

5,184 

373,248 

8.485 

4.160 

APPENDIX  II  159 


Square 

Cube 

No. 

Square 

Cube 

Root 

Root 

73 

5,329 

389,017 

8.544 

4.179 

74 

5,476 

405,224 

8.602 

4.198 

75 

5,625 

421,875 

8.660 

4.217 

76 

5,776 

438,976 

8.717 

4.235 

77 

5,929 

456,533 

8.774 

4.254 

78 

6,084 

474,552 

8.831 

4.272 

79 

6,241 

493,039 

8.888 

4.290 

80 

6,400 

512,000 

8.944 

4.308 

81 

6,561 

531,441 

9.000 

4.326 

82 

6,724 

551,368 

9.055 

4.344 

83 

6,889 

571,787 

9.110 

4.362 

84 

7,056 

592,704 

9.165 

4.379 

85 

7,225 

614,125 

9.219 

4.396 

86 

7,396 

636,056 

9.273 

4.414 

87 

7,569 

658,503 

9.327 

4.431 

88 

7,744 

681,472 

9.380 

4.447 

89 

7,921 

704,969 

9.433 

4.464 

90 

8,100 

729,000 

9.486 

4.481 

91 

8,281 

753,571 

9.539 

4.497 

92 

8,464 

778,688 

9.591 

4.514 

93 

8,649 

804,357 

9.643 

4.530 

94 

8,836 

830,584 

9.695 

4.546 

95 

9,025 

857,375 

9.746 

4.562 

96 

9,216 

884,736 

9.797 

4.578 

97 

9,409 

912,673 

9.848 

4.594 

98 

9,604 

941,192 

9.899 

4.610 

99 

9,801 

970,299 

9.949 

4.626 

100 

10,000 

1,000,000 

10.000 

4.641 

Note:   In  the  above  table  the  last  two  columns  are  correct  to  three 
decimal  places,  without  allowance  for  decimals  dropped. 


York 


1  r89 

2  »21 

3  [39 

5  193 

6  k02 

7  .91 

8  Ms 

9  m 

10  111 

11  54 

12  90 

13f75 

15 
16 
17 
18 
19 
20 
21 
22 

23  , 

24  57 


1850 

2,995,772 

23,191,876 

7.74 

7,135,780,000 

307.69 

63,452,774 

2.74 

63,452,774 

3,7:2,393 

0.16 

31,981,739 

1,866,100 

147,395,456 


131,366,526 

278,761,982 
12.02 


1840 

1,793,299 

17,069,453 

9 .  o2 


3,573,344 
0.21 

3,573,344 

174,698 

0.01 

1,675,483 

1,726,703 

79,336,916 


106,968,572 

186,305,488 

10.91 


1830 

1,793,299 

12,866,020 

7.17 


48,565,407 

3.77 

48,565,406 

1,912,575 

0.15 

643,105 

2,495,400 

26,344,295 


61,000,000 

87,344,295 

6.79 


1800 


843,246 

5,308,483 

0.30 


1 
2 
3 

4 

5 

82,976,294  6 

15.63  7 

82,976,294  8 

3,402,601  9 

0.64  10 

317,760  11 

224,296  12 

16,000,000  \^ 

15 

16 

17 

18 

10,600,000  19 

26,500,000  20 

5.00  21 

22 

23 

24 

25 


A  Picture  of  the  Progress  of  the  United  States  During  120  Years  of  National  Life 

CompUed  (roni  umdal  Sources  by  O.   F.  Austin.  StatisUclan  or  The  NaUooal  City   Bank  of   New  York 
in.  1910  1900  189U  188U  1870  1800 


K  i^ 


rcipts,    not   pr^loualf   I 


d  )lBS.e80.01T.  ropMtlrelr. 


b 


o 


INDEX 


Aggregate  method,  56 

Aldrich  Eeport,  4 

American    Economic    Review,    53, 

96,  124,  127 
American    Statistical    Association, 

96-148 
Annalist,  The,  97 
Annalist  Barometer,  125 
Arithmetic  mean,  20 
Average,  20 
Average  deviation,  36 

B 

Babson,  Roger  W.,  76,  124,  126 

Babson  's  index,  65 

Bailey,  W.  B.,  19 

Bowley,  Arthur  L.,  19,  44,  67,  148 

Bradstreet's  index,  64,  124 

Bureau  of  Labor  Statistics,  51,  56, 

63,   113,  124 
Barnett,  George  E.,  66 
Brinton,  W.  C,  44 
Business  barometers,  115 
Business  cycles,  123 
Business  statistics,  3 


Circulation,  98 
Concurrent  deviations,  132 
Continuous  series,  5 
Correlation,  131 
Cost  of  living,  53 
Corn,  76,  77 
Cummings,  John,  19 
Cycles,  101,  113,  137 


Day,  E.  E.,  19,  96 
Derived  table,  13 
Discrete  series,  5 
Dun's  index,  64 


E 
Exports,  129,  150 

F 

Falkner,  R.  P.,  3 

Farm  wages,  70 

Federal    Reserve    Bank    of    New 

York,    124 
Federal  Reserve  Board,  64 
Federal  Reserve  Bulletin,  65 
Field,  J.  A.,  44 
Fisher,  Irving,  44,  67,  96 
Fisher's  index,  86 
Food  prices,  56 
Foreign    exchange,    68 
Free-hand  method,  100 
Frequency  curve,  9 
Frequency  polygon,  16 

G 

Graphic  work,  153 


Hansen,  A.  H.,  124 
Howard,    Stanley   E.,    49,    67 
Hoffmann,  F.  L.,  96 
Hurlin,  Ralph  G.,  126 


Income  distribution,  92 

Indexes,  47,  62,   65 

Index  of  physical  production,  75 

Index  of  quantity,  74 

Index  of  value,  74 

Ingalls,  W.  R.,  96 

Interest  rates,  118 

Interpolation,  30 


Jevons,  Stanley  W.,  148 
Jordon,  D.  F.,  121,  126,  127 


161 


162 


INDEX 


Kemmerer,  E.  "W.,  88,  123,  127 
King,  W.  I.,  44,  96,  98 
Knauth,   Oswald  W.,  92 
Koren,  John,  19 


Lettering,  153 

Line  of  least  squares,  105,  135 
Link-relative   method,   120 
Lorenz  curve,  40,  44 

M 

Machine  tabulation,  18 

Macauley,  Frederick  E.,  92 

Marshall,  Wm.  C,  44 

Median,  the,  27 

Meeker,  Royal,  67,  96 

Method  of  rank-differences,  147 

Method  of  averages,  118 

Method  of  semi-averages,  102 

Method  of  standard  quantities,  83 

Mitchell,  Wesley  C,  67,  92,  127 

Mode,  21 

Monthly  Labor  Review,  29,  70 

Monthly  Review,  118,  123 

Moore,  H.  L.,  126,  127,  144 

Moving  average,  102 

N 

National  Bureau  of  Economic  Re- 
search, 91,  96 

National  income,  91 

National  Industrial  Conference 
Board,  54 

Normal  curve,  10 


0 


Ogive,  34,  36 


Primary  table,  5 
Probable  error,  30,  140 
Property  ownership,  96 
Proportional   expenditure   method. 

57 
Pareto's  law,  94 

Q 

Quantity  indexes,  74 
Quantity  theory,  87 
Quartile    deviation,   29 
Quartile  dispersion,  32 
Quarterly    Journal   of   Economics, 
109,  126 


Rank-differences,   147 

Ratio  paper,  33 

Real  wages,  48,  51,  53 

Rectangular  histogram,  25 

Reference  list,  156 

Review  of  Economic  Statistics,  96, 

121,   123,   124,  126 
Royal  Statistical  Society,  67 
Rugg,  H.  O.,  19 

S 

Salaries,  16 

Salaries  in  Universities,  28 
Schedules,  4 
Seasonal  index,  120 
Seasonal  variations,  116 
Secrist,  Horace,  19,  44,  67,  148 
Semi-logarithmic  paper,  33 
Skewness,  43 
Slichter,  C.  S.,  105 
Sources  of  data,  155 
Standard  price,  78 
Statistical  units,  17 
Stewart,  Walter  W.,  67,  96 


Parabola  trend,  110 

Pearson  method,  134 

Persons,  W.  W.,  121,  127,  148 

Peddle,  John  B.,  127 

Pig  iron,  76,  77,  128 

Physical  production  in  U.   S.,   80, 

81,  98 
Pratt,  Andrew  A.,  127 
Price  indexes,  theory  of,   82 
Prices,  51,  55,  60,  63,  65,  77,  113, 

128,  142  »       »         > 


Tabulation  methods,  18 
Tally  sheet,  11 
Tingley,  Richard  H.,  127 
Trends,  nature  of,  100 
Types  of  graphs,  154 
Types  compared,  27 


Value  indexes,  74 

Visible  supply  of  wheat,  141,  150 


INDEX  163 

W  Weighted  average,  20 

Weights,  cost  of  living,  71 
Wages  and  salaries,  16  West,  Carl  S.,  148 

Wage  indexes,  49,  51,  53,  70  Wheat,  76,  77,  129,  141,  150 

Wage  roll,  7  Whipple,  G.  C,  44 

Walsh,  C.  M.,  97  Working,  Holbrook,  97 


r^ov 


This  book  is  DUE  on  the  last  date  stamped  below 

BEC'D  URL  CiRC 

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OCT  1  9  1950 


Form  L-9-10m-5,'28 


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r 


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ERSITY  of  CALIFORNIA, 
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